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The Algebra and Geometry of VectorsMath 218
Brian D. Fitzpatrick
Duke University
November 1, 2019
MATH
Overview
MotivationWhat We KnowWhat We Want
Vectors in R2
Geometric InterpretationTails and TipsCoordinatesLength
Vectors in R3
Geometric InterpretationLength
Vectors in Rn
“Geometric” InterpretationLength
Scalar-Vector MultiplicationGeometric InterpretationAlgebraic ComparisonUnit VectorsNormalization
Vector AdditionGeometric InterpretationProperties
The Dot ProductDefinitionPropertiesLengthsAngles
MotivationWhat We Know
So far, the vector operations we have learned are
scalar-vector muliplication c · #»v vector addition #»v + #»w
These operations are algebraic
c ·
v1...vn
=
c · v1...
c · vn
v1
...vn
+
w1...wn
=
v1 + w1...
vn + wn
MotivationWhat We Know
So far, the vector operations we have learned are
scalar-vector muliplication c · #»v vector addition #»v + #»w
These operations are algebraic
c ·
v1...vn
=
c · v1...
c · vn
v1
...vn
+
w1...wn
=
v1 + w1...
vn + wn
MotivationWhat We Know
So far, the vector operations we have learned are
scalar-vector muliplication c · #»v vector addition #»v + #»w
These operations are algebraic
c ·
v1...vn
=
c · v1...
c · vn
v1...vn
+
w1...wn
=
v1 + w1...
vn + wn
MotivationWhat We Know
So far, the vector operations we have learned are
scalar-vector muliplication c · #»v vector addition #»v + #»w
These operations are algebraic
c ·
v1...vn
=
c · v1...
c · vn
v1
...vn
+
w1...wn
=
v1 + w1...
vn + wn
MotivationWhat We Want
Can we interpret these operations geometrically?
Vectors in R2
Geometric Interpretation
Geometrically, a vector in R2 is represented by an arrow in thexy -plane.
x
y
#»a
#»
b
#»v
#»w
#»x#»y
#»z
Vectors in R2
Geometric Interpretation
Geometrically, a vector in R2 is represented by an arrow in thexy -plane.
x
y
#»a
#»
b
#»v
#»w
#»x#»y
#»z
Vectors in R2
Geometric Interpretation
Geometrically, a vector in R2 is represented by an arrow in thexy -plane.
x
y
#»a
#»
b
#»v
#»w
#»x#»y
#»z
Vectors in R2
Geometric Interpretation
Geometrically, a vector in R2 is represented by an arrow in thexy -plane.
x
y
#»a
#»
b
#»v
#»w
#»x#»y
#»z
Vectors in R2
Geometric Interpretation
Geometrically, a vector in R2 is represented by an arrow in thexy -plane.
x
y
#»a
#»
b
#»v
#»w
#»x#»y
#»z
Vectors in R2
Geometric Interpretation
Geometrically, a vector in R2 is represented by an arrow in thexy -plane.
x
y
#»a
#»
b
#»v
#»w
#»x#»y
#»z
Vectors in R2
Geometric Interpretation
Geometrically, a vector in R2 is represented by an arrow in thexy -plane.
x
y
#»a
#»
b
#»v
#»w
#»x
#»y
#»z
Vectors in R2
Geometric Interpretation
Geometrically, a vector in R2 is represented by an arrow in thexy -plane.
x
y
#»a
#»
b
#»v
#»w
#»x#»y
#»z
Vectors in R2
Geometric Interpretation
Geometrically, a vector in R2 is represented by an arrow in thexy -plane.
x
y
#»a
#»
b
#»v
#»w
#»x#»y
#»z
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v
=# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w
=# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v
=# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w
=# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v
=# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w
=# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v
=# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w
=# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v
=# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w
=# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w
=# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w
=# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w
=# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w
=# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Tails and Tips
Every arrow emanates from a tail and terminates at a tip.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =# »
PQ
•P(−3,−1)
“tail”
•Q(2, 2)“tip”
#»w =# »
RS
•R(−2,−2)
“tail”
•S(3,−1)
“tip”
Vectors in R2
Coordinates
The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.
v1 =
x1 − x0
v2 =
y1 − y0
#»v=〈v1, v2〉
•
•
(x0, y0)
(x1, y1)
Vectors in R2
Coordinates
The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.
v1 =
x1 − x0
v2 =
y1 − y0
#»v=〈v1, v2〉
•
•
(x0, y0)
(x1, y1)
Vectors in R2
Coordinates
The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.
v1 =
x1 − x0
v2 =
y1 − y0
#»v=〈v1, v2〉
•
•
(x0, y0)
(x1, y1)
Vectors in R2
Coordinates
The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.
v1 = x1 − x0
v2 =
y1 − y0
#»v=〈v1, v2〉
•
•
(x0, y0)
(x1, y1)
Vectors in R2
Coordinates
The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.
v1 = x1 − x0
v2 = y1 − y0#»v=〈v1, v2〉
•
•
(x0, y0)
(x1, y1)
Vectors in R2
Coordinates
x
y
−3 −2 −1 1 2 3
−2
−1
1
2#»v = 〈2− (−3), 2− (−1)〉 = 〈5, 3〉
•P(−3,−1)
•Q(2, 2)
#»w = 〈3− (−2),−1− (−2)〉 = 〈5, 1〉
•R(−2,−2)
•S(3,−1)
Vectors in R2
Coordinates
Two arrows define the same vector if they have the samecoordinates.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =
〈−2, −1〉
#»w =
〈−2, −1〉
#»v = #»w
Vectors in R2
Coordinates
Two arrows define the same vector if they have the samecoordinates.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =
〈−2, −1〉
#»w =
〈−2, −1〉
#»v = #»w
Vectors in R2
Coordinates
Two arrows define the same vector if they have the samecoordinates.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =
〈−2, −1〉
#»w =
〈−2, −1〉
#»v = #»w
Vectors in R2
Coordinates
Two arrows define the same vector if they have the samecoordinates.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =〈−2, −
1〉
#»w =
〈−2, −1〉
#»v = #»w
Vectors in R2
Coordinates
Two arrows define the same vector if they have the samecoordinates.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =〈−2, −
1〉
#»w =
〈−2, −1〉
#»v = #»w
Vectors in R2
Coordinates
Two arrows define the same vector if they have the samecoordinates.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =〈−2, −
1〉
#»w =〈−2, −
1〉
#»v = #»w
Vectors in R2
Coordinates
Two arrows define the same vector if they have the samecoordinates.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v =〈−2, −
1〉
#»w =〈−2, −
1〉#»v = #»w
Vectors in R2
Coordinates
Without context, it is convention to plot vectors using the origin asthe tail.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v = 〈3, 2〉
#»w = 〈−3, 1〉
#»x = 〈−2,−2〉
Vectors in R2
Coordinates
Without context, it is convention to plot vectors using the origin asthe tail.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v = 〈3, 2〉
#»w = 〈−3, 1〉
#»x = 〈−2,−2〉
Vectors in R2
Coordinates
Without context, it is convention to plot vectors using the origin asthe tail.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v = 〈3, 2〉
#»w = 〈−3, 1〉
#»x = 〈−2,−2〉
Vectors in R2
Coordinates
Without context, it is convention to plot vectors using the origin asthe tail.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v = 〈3, 2〉
#»w = 〈−3, 1〉
#»x = 〈−2,−2〉
Vectors in R2
Coordinates
Without context, it is convention to plot vectors using the origin asthe tail.
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
#»v = 〈3, 2〉
#»w = 〈−3, 1〉
#»x = 〈−2,−2〉
Vectors in R2
Length
The length of a vector #»v = 〈v1, v2〉 is ‖ #»v ‖ =
√v21 + v22 .
|v1|
|v2|‖ #»v ‖
Vectors in R2
Length
The length of a vector #»v = 〈v1, v2〉 is ‖ #»v ‖ =
√v21 + v22 .
|v1|
|v2|
‖ #»v ‖
Vectors in R2
Length
The length of a vector #»v = 〈v1, v2〉 is ‖ #»v ‖ =
√v21 + v22 .
|v1|
|v2|‖ #»v ‖
Vectors in R2
Length
The length of a vector #»v = 〈v1, v2〉 is ‖ #»v ‖ =√
v21 + v22 .
|v1|
|v2|‖ #»v ‖
Vectors in R2
Length
Example
The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are
‖ #»r ‖ =√
(1)2 + (−1)2 ‖ #»p ‖ =√
(3)2 + (2)2
=√
1 + 1 =√
9 + 4
=√
2 =√
13
Vectors in R2
Length
Example
The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are
‖ #»r ‖ =√
(1)2 + (−1)2
‖ #»p ‖ =√
(3)2 + (2)2
=√
1 + 1 =√
9 + 4
=√
2 =√
13
Vectors in R2
Length
Example
The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are
‖ #»r ‖ =√
(1)2 + (−1)2
‖ #»p ‖ =√
(3)2 + (2)2
=√
1 + 1
=√
9 + 4
=√
2 =√
13
Vectors in R2
Length
Example
The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are
‖ #»r ‖ =√
(1)2 + (−1)2
‖ #»p ‖ =√
(3)2 + (2)2
=√
1 + 1
=√
9 + 4
=√
2
=√
13
Vectors in R2
Length
Example
The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are
‖ #»r ‖ =√
(1)2 + (−1)2 ‖ #»p ‖ =√
(3)2 + (2)2
=√
1 + 1
=√
9 + 4
=√
2
=√
13
Vectors in R2
Length
Example
The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are
‖ #»r ‖ =√
(1)2 + (−1)2 ‖ #»p ‖ =√
(3)2 + (2)2
=√
1 + 1 =√
9 + 4
=√
2
=√
13
Vectors in R2
Length
Example
The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are
‖ #»r ‖ =√
(1)2 + (−1)2 ‖ #»p ‖ =√
(3)2 + (2)2
=√
1 + 1 =√
9 + 4
=√
2 =√
13
Vectors in R2
Length
Other Terms for “Length”
“norm” “magnitude”
Alternate Notation for ‖ #»v ‖Some people write | #»v | instead of ‖ #»v ‖.
Vectors in R2
Length
Other Terms for “Length”
“norm” “magnitude”
Alternate Notation for ‖ #»v ‖Some people write | #»v | instead of ‖ #»v ‖.
Vectors in R3
Geometric Interpretation
We think of vectors in R3 as objects in xyz-space.
x
y
z
3
3
−4
• P(3, 3,−4)
−4
4
5
• Q(−4, 4, 5)
Vectors in R3
Geometric Interpretation
We think of vectors in R3 as objects in xyz-space.
x
y
z
3
3
−4
• P(3, 3,−4)
−4
4
5
• Q(−4, 4, 5)
Vectors in R3
Geometric Interpretation
We think of vectors in R3 as objects in xyz-space.
x
y
z
3
3
−4
• P(3, 3,−4)
−4
4
5
• Q(−4, 4, 5)
Vectors in R3
Geometric Interpretation
Geometrically, a vector in R3 is represented by an arrow inxyz-space.
x
y
z
P(11, 2, 0)
Q(8,−4, 6)
R(−2, 2, 7)
S(1, 8, 1)
#»v
#»w
#»x
#»v =
−2− 82− (−4)
7− 6
=
−1061
#»w =
−2− 112− 27− 0
=
−1307
#»x =
1− 118− 21− 0
=
−1061
Note that #»v = #»x !
Vectors in R3
Geometric Interpretation
Geometrically, a vector in R3 is represented by an arrow inxyz-space.
x
y
z
P(11, 2, 0)
Q(8,−4, 6)
R(−2, 2, 7)
S(1, 8, 1)
#»v
#»w
#»x
#»v =
−2− 82− (−4)
7− 6
=
−1061
#»w =
−2− 112− 27− 0
=
−1307
#»x =
1− 118− 21− 0
=
−1061
Note that #»v = #»x !
Vectors in R3
Geometric Interpretation
Geometrically, a vector in R3 is represented by an arrow inxyz-space.
x
y
z
P(11, 2, 0)
Q(8,−4, 6)
R(−2, 2, 7)
S(1, 8, 1)
#»v
#»w
#»x
#»v =
−2− 82− (−4)
7− 6
=
−1061
#»w =
−2− 112− 27− 0
=
−1307
#»x =
1− 118− 21− 0
=
−1061
Note that #»v = #»x !
Vectors in R3
Geometric Interpretation
Geometrically, a vector in R3 is represented by an arrow inxyz-space.
x
y
z
P(11, 2, 0)
Q(8,−4, 6)
R(−2, 2, 7)
S(1, 8, 1)
#»v
#»w
#»x
#»v =
−2− 82− (−4)
7− 6
=
−1061
#»w =
−2− 112− 27− 0
=
−1307
#»x =
1− 118− 21− 0
=
−1061
Note that #»v = #»x !
Vectors in R3
Geometric Interpretation
Geometrically, a vector in R3 is represented by an arrow inxyz-space.
x
y
z
P(11, 2, 0)
Q(8,−4, 6)
R(−2, 2, 7)
S(1, 8, 1)
#»v
#»w
#»x
#»v =
−2− 82− (−4)
7− 6
=
−1061
#»w =
−2− 112− 27− 0
=
−1307
#»x =
1− 118− 21− 0
=
−1061
Note that #»v = #»x !
Vectors in R3
Geometric Interpretation
Geometrically, a vector in R3 is represented by an arrow inxyz-space.
x
y
z
P(11, 2, 0)
Q(8,−4, 6)
R(−2, 2, 7)
S(1, 8, 1)
#»v
#»w
#»x
#»v =
−2− 82− (−4)
7− 6
=
−1061
#»w =
−2− 112− 27− 0
=
−1307
#»x =
1− 118− 21− 0
=
−1061
Note that #»v = #»x !
Vectors in R3
Geometric Interpretation
Geometrically, a vector in R3 is represented by an arrow inxyz-space.
x
y
z
P(11, 2, 0)
Q(8,−4, 6)
R(−2, 2, 7)
S(1, 8, 1)
#»v
#»w
#»x
#»v =
−2− 82− (−4)
7− 6
=
−1061
#»w =
−2− 112− 27− 0
=
−1307
#»x =
1− 118− 21− 0
=
−1061
Note that #»v = #»x !
Vectors in R3
Geometric Interpretation
Geometrically, a vector in R3 is represented by an arrow inxyz-space.
x
y
z
P(11, 2, 0)
Q(8,−4, 6)
R(−2, 2, 7)
S(1, 8, 1)
#»v
#»w
#»x
#»v =
−2− 82− (−4)
7− 6
=
−1061
#»w =
−2− 112− 27− 0
=
−1307
#»x =
1− 118− 21− 0
=
−1061
Note that #»v = #»x !
Vectors in R3
Length
DefinitionThe length of #»v = 〈v1, v2, v3〉 is ‖ #»v ‖ =
√v21 + v22 + v23 .
Example
The length of #»z = 〈−7, −8, 5〉 is
‖ #»z ‖ =√
(−7)2 + (−8)2 + (5)2
=√
49 + 64 + 25
=√
138
Vectors in R3
Length
DefinitionThe length of #»v = 〈v1, v2, v3〉 is ‖ #»v ‖ =
√v21 + v22 + v23 .
Example
The length of #»z = 〈−7, −8, 5〉 is
‖ #»z ‖ =√
(−7)2 + (−8)2 + (5)2
=√
49 + 64 + 25
=√
138
Vectors in Rn
“Geometric” Interpretation
The “visible” geometry of R2 and R3 is used to define geometry inhigher dimensions.
#»a
#»
b
#»v
#»w
#»x#»y
#»z
We think of a vector #»v ∈ Rn as an “arrow” in Euclidean n-space.
Vectors in Rn
“Geometric” Interpretation
The “visible” geometry of R2 and R3 is used to define geometry inhigher dimensions.
#»a
#»
b
#»v
#»w
#»x#»y
#»z
We think of a vector #»v ∈ Rn as an “arrow” in Euclidean n-space.
Vectors in Rn
“Geometric” Interpretation
The “visible” geometry of R2 and R3 is used to define geometry inhigher dimensions.
#»a
#»
b
#»v
#»w
#»x#»y
#»z
We think of a vector #»v ∈ Rn as an “arrow” in Euclidean n-space.
Vectors in Rn
Length
The length of #»v = 〈v1, v2, . . . , vn〉 is
‖ #»v ‖ =
√v21 + v22 + · · ·+ v2n
Even if we can’t “see” a vector, we can still compute its length.
Vectors in Rn
Length
The length of #»v = 〈v1, v2, . . . , vn〉 is
‖ #»v ‖ =√v21 + v22 + · · ·+ v2n
Even if we can’t “see” a vector, we can still compute its length.
Vectors in Rn
Length
The length of #»v = 〈v1, v2, . . . , vn〉 is
‖ #»v ‖ =√v21 + v22 + · · ·+ v2n
Even if we can’t “see” a vector, we can still compute its length.
Vectors in Rn
Length
Example
The length of #»v = 〈4, 0, −2, −5, −1〉 is
‖ #»v ‖ =√
(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2
=√
16 + 0 + 4 + 25 + 1
=√
46
Example
Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈ R40 and the length of #»a is ‖ #»a ‖ =
√2.
Vectors in Rn
Length
Example
The length of #»v = 〈4, 0, −2, −5, −1〉 is
‖ #»v ‖ =√
(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2
=√
16 + 0 + 4 + 25 + 1
=√
46
Example
Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈
R40 and the length of #»a is ‖ #»a ‖ =√
2.
Vectors in Rn
Length
Example
The length of #»v = 〈4, 0, −2, −5, −1〉 is
‖ #»v ‖ =√
(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2
=√
16 + 0 + 4 + 25 + 1
=√
46
Example
Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈ R40
and the length of #»a is ‖ #»a ‖ =√
2.
Vectors in Rn
Length
Example
The length of #»v = 〈4, 0, −2, −5, −1〉 is
‖ #»v ‖ =√
(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2
=√
16 + 0 + 4 + 25 + 1
=√
46
Example
Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈ R40 and the length of #»a is ‖ #»a ‖ =
√2.
Vectors in Rn
Length
Example
The length of #»v = 〈4, 0, −2, −5, −1〉 is
‖ #»v ‖ =√
(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2
=√
16 + 0 + 4 + 25 + 1
=√
46
Example
Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈ R40 and the length of #»a is ‖ #»a ‖ =
√2.
Scalar-Vector MultiplicationGeometric Interpretation
QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?
#»v2·#»v
(1/2
) ·#»v
(−1/2
) ·#»v
−1·#»v
Definition (Geometric)
The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.
Scalar-Vector MultiplicationGeometric Interpretation
QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?
#»v
2·#»v
(1/2
) ·#»v
(−1/2
) ·#»v
−1·#»v
Definition (Geometric)
The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.
Scalar-Vector MultiplicationGeometric Interpretation
QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?
#»v2·#»v
(1/2
) ·#»v
(−1/2
) ·#»v
−1·#»v
Definition (Geometric)
The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.
Scalar-Vector MultiplicationGeometric Interpretation
QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?
#»v2·#»v
(1/2
) ·#»v
(−1/2
) ·#»v
−1·#»v
Definition (Geometric)
The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.
Scalar-Vector MultiplicationGeometric Interpretation
QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?
#»v2·#»v
(1/2
) ·#»v
(−1/2
) ·#»v
−1·#»v
Definition (Geometric)
The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.
Scalar-Vector MultiplicationGeometric Interpretation
QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?
#»v2·#»v
(1/2
) ·#»v
(−1/2
) ·#»v
−1·#»v
Definition (Geometric)
The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.
Scalar-Vector MultiplicationGeometric Interpretation
QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?
#»v2·#»v
(1/2
) ·#»v
(−1/2
) ·#»v
−1·#»v
Definition (Geometric)
The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.
Scalar-Vector MultiplicationAlgebraic Comparison
This “geometric” interpretation coincides with our “algebraic”definition.
‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖
=√
(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2
=√
c2 · v21 + c2 · v22 + · · ·+ c2 · v2n
=√
c2 · (v21 + v22 + · · ·+ v2n )
=√c2 ·
√v21 + v22 + · · ·+ v2n
= |c | · ‖ #»v ‖
Scalar-Vector MultiplicationAlgebraic Comparison
This “geometric” interpretation coincides with our “algebraic”definition.
‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖
=√
(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2
=√
c2 · v21 + c2 · v22 + · · ·+ c2 · v2n
=√
c2 · (v21 + v22 + · · ·+ v2n )
=√c2 ·
√v21 + v22 + · · ·+ v2n
= |c | · ‖ #»v ‖
Scalar-Vector MultiplicationAlgebraic Comparison
This “geometric” interpretation coincides with our “algebraic”definition.
‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖
=√
(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2
=√
c2 · v21 + c2 · v22 + · · ·+ c2 · v2n
=√
c2 · (v21 + v22 + · · ·+ v2n )
=√c2 ·
√v21 + v22 + · · ·+ v2n
= |c | · ‖ #»v ‖
Scalar-Vector MultiplicationAlgebraic Comparison
This “geometric” interpretation coincides with our “algebraic”definition.
‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖
=√
(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2
=√
c2 · v21 + c2 · v22 + · · ·+ c2 · v2n
=√
c2 · (v21 + v22 + · · ·+ v2n )
=√c2 ·
√v21 + v22 + · · ·+ v2n
= |c | · ‖ #»v ‖
Scalar-Vector MultiplicationAlgebraic Comparison
This “geometric” interpretation coincides with our “algebraic”definition.
‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖
=√
(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2
=√
c2 · v21 + c2 · v22 + · · ·+ c2 · v2n
=√
c2 · (v21 + v22 + · · ·+ v2n )
=√c2 ·
√v21 + v22 + · · ·+ v2n
= |c | · ‖ #»v ‖
Scalar-Vector MultiplicationAlgebraic Comparison
This “geometric” interpretation coincides with our “algebraic”definition.
‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖
=√
(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2
=√
c2 · v21 + c2 · v22 + · · ·+ c2 · v2n
=√
c2 · (v21 + v22 + · · ·+ v2n )
=√c2 ·
√v21 + v22 + · · ·+ v2n
= |c | · ‖ #»v ‖
Scalar-Vector MultiplicationUnit Vectors
DefinitionA unit vector is a vector with length one.
Example
Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then
‖ #»u ‖
= ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖
= (1/2) ·√
(1)2 + (−1)2 + (−1)2 + (1)2
= (1/2) ·√
1 + 1 + 1 + 1
= (1/2) ·√
4
= (1/2) · 2= 1
Scalar-Vector MultiplicationUnit Vectors
DefinitionA unit vector is a vector with length one.
Example
Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then
‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖
= |1/2| · ‖〈1, −1, −1, 1〉 ‖
= (1/2) ·√
(1)2 + (−1)2 + (−1)2 + (1)2
= (1/2) ·√
1 + 1 + 1 + 1
= (1/2) ·√
4
= (1/2) · 2= 1
Scalar-Vector MultiplicationUnit Vectors
DefinitionA unit vector is a vector with length one.
Example
Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then
‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖
= (1/2) ·√
(1)2 + (−1)2 + (−1)2 + (1)2
= (1/2) ·√
1 + 1 + 1 + 1
= (1/2) ·√
4
= (1/2) · 2= 1
Scalar-Vector MultiplicationUnit Vectors
DefinitionA unit vector is a vector with length one.
Example
Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then
‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖
= (1/2) ·√
(1)2 + (−1)2 + (−1)2 + (1)2
= (1/2) ·√
1 + 1 + 1 + 1
= (1/2) ·√
4
= (1/2) · 2= 1
Scalar-Vector MultiplicationUnit Vectors
DefinitionA unit vector is a vector with length one.
Example
Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then
‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖
= (1/2) ·√
(1)2 + (−1)2 + (−1)2 + (1)2
= (1/2) ·√
1 + 1 + 1 + 1
= (1/2) ·√
4
= (1/2) · 2= 1
Scalar-Vector MultiplicationUnit Vectors
DefinitionA unit vector is a vector with length one.
Example
Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then
‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖
= (1/2) ·√
(1)2 + (−1)2 + (−1)2 + (1)2
= (1/2) ·√
1 + 1 + 1 + 1
= (1/2) ·√
4
= (1/2) · 2= 1
Scalar-Vector MultiplicationUnit Vectors
DefinitionA unit vector is a vector with length one.
Example
Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then
‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖
= (1/2) ·√
(1)2 + (−1)2 + (−1)2 + (1)2
= (1/2) ·√
1 + 1 + 1 + 1
= (1/2) ·√
4
= (1/2) · 2
= 1
Scalar-Vector MultiplicationUnit Vectors
DefinitionA unit vector is a vector with length one.
Example
Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then
‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖
= (1/2) ·√
(1)2 + (−1)2 + (−1)2 + (1)2
= (1/2) ·√
1 + 1 + 1 + 1
= (1/2) ·√
4
= (1/2) · 2= 1
Scalar-Vector MultiplicationUnit Vectors
Example
Consider the vectors
#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉
Each #»e 1, #»e 2, and #»e 3 is a unit vector.
DefinitionThese vectors form the standard basis of R3.
Important
Every vector #»v = 〈v1, v2, v3〉 satisfies
#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3
Every vector is a linear combination of the standard basis!
Scalar-Vector MultiplicationUnit Vectors
Example
Consider the vectors
#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉
Each #»e 1, #»e 2, and #»e 3 is a unit vector.
DefinitionThese vectors form the standard basis of R3.
Important
Every vector #»v = 〈v1, v2, v3〉 satisfies
#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3
Every vector is a linear combination of the standard basis!
Scalar-Vector MultiplicationUnit Vectors
Example
Consider the vectors
#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉
Each #»e 1, #»e 2, and #»e 3 is a unit vector.
DefinitionThese vectors form the standard basis of R3.
Important
Every vector #»v = 〈v1, v2, v3〉 satisfies
#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3
Every vector is a linear combination of the standard basis!
Scalar-Vector MultiplicationUnit Vectors
Example
Consider the vectors
#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉
Each #»e 1, #»e 2, and #»e 3 is a unit vector.
DefinitionThese vectors form the standard basis of R3.
Important
Every vector #»v = 〈v1, v2, v3〉 satisfies
#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3
Every vector is a
linear combination of the standard basis!
Scalar-Vector MultiplicationUnit Vectors
Example
Consider the vectors
#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉
Each #»e 1, #»e 2, and #»e 3 is a unit vector.
DefinitionThese vectors form the standard basis of R3.
Important
Every vector #»v = 〈v1, v2, v3〉 satisfies
#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3
Every vector is a linear combination of the standard basis!
Scalar-Vector MultiplicationNormalization
DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .
#»v
v̂
Note that
‖v̂‖ =
∥∥∥∥ 1
‖ #»v ‖· #»v
∥∥∥∥ =
∣∣∣∣ 1
‖ #»v ‖
∣∣∣∣ · ‖ #»v ‖ =1
‖ #»v ‖· ‖ #»v ‖ = 1
The normalization v̂ of #»v is always a unit vector!
Scalar-Vector MultiplicationNormalization
DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .
#»v
v̂
Note that
‖v̂‖ =
∥∥∥∥ 1
‖ #»v ‖· #»v
∥∥∥∥ =
∣∣∣∣ 1
‖ #»v ‖
∣∣∣∣ · ‖ #»v ‖ =1
‖ #»v ‖· ‖ #»v ‖ = 1
The normalization v̂ of #»v is always a unit vector!
Scalar-Vector MultiplicationNormalization
DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .
#»v
v̂
Note that
‖v̂‖ =
∥∥∥∥ 1
‖ #»v ‖· #»v
∥∥∥∥ =
∣∣∣∣ 1
‖ #»v ‖
∣∣∣∣ · ‖ #»v ‖ =1
‖ #»v ‖· ‖ #»v ‖ = 1
The normalization v̂ of #»v is always a unit vector!
Scalar-Vector MultiplicationNormalization
DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .
#»v
v̂
Note that
‖v̂‖ =
∥∥∥∥ 1
‖ #»v ‖· #»v
∥∥∥∥ =
∣∣∣∣ 1
‖ #»v ‖
∣∣∣∣ · ‖ #»v ‖ =1
‖ #»v ‖· ‖ #»v ‖ = 1
The normalization v̂ of #»v is always a unit vector!
Scalar-Vector MultiplicationNormalization
DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .
#»v
v̂
Note that
‖v̂‖ =
∥∥∥∥ 1
‖ #»v ‖· #»v
∥∥∥∥ =
∣∣∣∣ 1
‖ #»v ‖
∣∣∣∣ · ‖ #»v ‖ =
1
‖ #»v ‖· ‖ #»v ‖ = 1
The normalization v̂ of #»v is always a unit vector!
Scalar-Vector MultiplicationNormalization
DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .
#»v
v̂
Note that
‖v̂‖ =
∥∥∥∥ 1
‖ #»v ‖· #»v
∥∥∥∥ =
∣∣∣∣ 1
‖ #»v ‖
∣∣∣∣ · ‖ #»v ‖ =1
‖ #»v ‖· ‖ #»v ‖ =
1
The normalization v̂ of #»v is always a unit vector!
Scalar-Vector MultiplicationNormalization
DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .
#»v
v̂
Note that
‖v̂‖ =
∥∥∥∥ 1
‖ #»v ‖· #»v
∥∥∥∥ =
∣∣∣∣ 1
‖ #»v ‖
∣∣∣∣ · ‖ #»v ‖ =1
‖ #»v ‖· ‖ #»v ‖ = 1
The normalization v̂ of #»v is always a unit vector!
Scalar-Vector MultiplicationNormalization
DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .
#»v
v̂
Note that
‖v̂‖ =
∥∥∥∥ 1
‖ #»v ‖· #»v
∥∥∥∥ =
∣∣∣∣ 1
‖ #»v ‖
∣∣∣∣ · ‖ #»v ‖ =1
‖ #»v ‖· ‖ #»v ‖ = 1
The normalization v̂ of #»v is always a unit vector!
Vector AdditionGeometric Interpretation
To interpret #»v + #»w geometrically, start by forming a parallelogram.
The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .
#»v
#»w
#»v
#»w#»v
+#»w
Vector AdditionGeometric Interpretation
To interpret #»v + #»w geometrically, start by forming a parallelogram.
The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .
#»v
#»w
#»v
#»w#»v
+#»w
Vector AdditionGeometric Interpretation
To interpret #»v + #»w geometrically, start by forming a parallelogram.
The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .
#»v
#»w
#»v
#»w#»v
+#»w
Vector AdditionGeometric Interpretation
To interpret #»v + #»w geometrically, start by forming a parallelogram.
The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .
#»v
#»w
#»v
#»w
#»v+
#»w
Vector AdditionGeometric Interpretation
To interpret #»v + #»w geometrically, start by forming a parallelogram.The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .
#»v
#»w
#»v
#»w
#»v+
#»w
Vector AdditionGeometric Interpretation
To interpret #»v + #»w geometrically, start by forming a parallelogram.The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .
#»v
#»w
#»v
#»w#»v
+#»w
Vector AdditionGeometric Interpretation
This geometric interpretation allows us to construct vectordiagrams.
#»v#»w#»x
#»y =
#»v − #»x
#»z =
#»w − #»v
Vector AdditionGeometric Interpretation
This geometric interpretation allows us to construct vectordiagrams.
#»v#»w#»x
#»y = #»v − #»x
#»z =
#»w − #»v
Vector AdditionGeometric Interpretation
This geometric interpretation allows us to construct vectordiagrams.
#»v#»w#»x
#»y = #»v − #»x
#»z = #»w − #»v
Vector AdditionProperties
Properties of Vector Arithmetic
Vector arithmetic obeys the following laws.
additive commutativity #»v + #»w = #»w + #»v
additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )
multiplicitive associativity (c · d) · #»v = c · (d · #»v )
scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w
vector distribution (c + d) · #»v = c · #»v + d · #»v
additive identity the zero vector#»
O = 〈0, 0, . . . , 0〉satisfies the equation
#»
O + #»v = #»v
NoteThe zero vector
#»
O is the only vector whose length is zero ‖ #»
O‖ = 0.
Vector AdditionProperties
Properties of Vector Arithmetic
Vector arithmetic obeys the following laws.
additive commutativity #»v + #»w = #»w + #»v
additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )
multiplicitive associativity (c · d) · #»v = c · (d · #»v )
scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w
vector distribution (c + d) · #»v = c · #»v + d · #»v
additive identity the zero vector#»
O = 〈0, 0, . . . , 0〉satisfies the equation
#»
O + #»v = #»v
NoteThe zero vector
#»
O is the only vector whose length is zero ‖ #»
O‖ = 0.
Vector AdditionProperties
Properties of Vector Arithmetic
Vector arithmetic obeys the following laws.
additive commutativity #»v + #»w = #»w + #»v
additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )
multiplicitive associativity (c · d) · #»v = c · (d · #»v )
scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w
vector distribution (c + d) · #»v = c · #»v + d · #»v
additive identity the zero vector#»
O = 〈0, 0, . . . , 0〉satisfies the equation
#»
O + #»v = #»v
NoteThe zero vector
#»
O is the only vector whose length is zero ‖ #»
O‖ = 0.
Vector AdditionProperties
Properties of Vector Arithmetic
Vector arithmetic obeys the following laws.
additive commutativity #»v + #»w = #»w + #»v
additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )
multiplicitive associativity (c · d) · #»v = c · (d · #»v )
scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w
vector distribution (c + d) · #»v = c · #»v + d · #»v
additive identity the zero vector#»
O = 〈0, 0, . . . , 0〉satisfies the equation
#»
O + #»v = #»v
NoteThe zero vector
#»
O is the only vector whose length is zero ‖ #»
O‖ = 0.
Vector AdditionProperties
Properties of Vector Arithmetic
Vector arithmetic obeys the following laws.
additive commutativity #»v + #»w = #»w + #»v
additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )
multiplicitive associativity (c · d) · #»v = c · (d · #»v )
scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w
vector distribution (c + d) · #»v = c · #»v + d · #»v
additive identity the zero vector#»
O = 〈0, 0, . . . , 0〉satisfies the equation
#»
O + #»v = #»v
NoteThe zero vector
#»
O is the only vector whose length is zero ‖ #»
O‖ = 0.
Vector AdditionProperties
Properties of Vector Arithmetic
Vector arithmetic obeys the following laws.
additive commutativity #»v + #»w = #»w + #»v
additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )
multiplicitive associativity (c · d) · #»v = c · (d · #»v )
scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w
vector distribution (c + d) · #»v = c · #»v + d · #»v
additive identity the zero vector#»
O = 〈0, 0, . . . , 0〉satisfies the equation
#»
O + #»v = #»v
NoteThe zero vector
#»
O is the only vector whose length is zero ‖ #»
O‖ = 0.
Vector AdditionProperties
Properties of Vector Arithmetic
Vector arithmetic obeys the following laws.
additive commutativity #»v + #»w = #»w + #»v
additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )
multiplicitive associativity (c · d) · #»v = c · (d · #»v )
scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w
vector distribution (c + d) · #»v = c · #»v + d · #»v
additive identity the zero vector#»
O = 〈0, 0, . . . , 0〉satisfies the equation
#»
O + #»v = #»v
NoteThe zero vector
#»
O is the only vector whose length is zero ‖ #»
O‖ = 0.
The Dot ProductDefinition
QuestionWhat does it mean to multiply two vectors?
AnswerThere is no correct answer. There are several useful ways tomultiply vectors.
The Dot ProductDefinition
QuestionWhat does it mean to multiply two vectors?
AnswerThere is no correct answer. There are several useful ways tomultiply vectors.
The Dot ProductDefinition
DefinitionThe dot product of #»v = 〈v1, v2, . . . , vn〉 and #»w = 〈w1,w2, . . . ,wn〉is defined as
#»v · #»w = v1 · w1 + v2 · w2 + · · ·+ vn · wn
NoteThe dot product of two vectors is a scalar, not a vector.
NoteThe dot product #»v · #»w is only defined if #»v and #»w have the samenumber of coordinates.
The Dot ProductDefinition
DefinitionThe dot product of #»v = 〈v1, v2, . . . , vn〉 and #»w = 〈w1,w2, . . . ,wn〉is defined as
#»v · #»w = v1 · w1 + v2 · w2 + · · ·+ vn · wn
NoteThe dot product of two vectors is a scalar, not a vector.
NoteThe dot product #»v · #»w is only defined if #»v and #»w have the samenumber of coordinates.
The Dot ProductDefinition
DefinitionThe dot product of #»v = 〈v1, v2, . . . , vn〉 and #»w = 〈w1,w2, . . . ,wn〉is defined as
#»v · #»w = v1 · w1 + v2 · w2 + · · ·+ vn · wn
NoteThe dot product of two vectors is a scalar, not a vector.
NoteThe dot product #»v · #»w is only defined if #»v and #»w have the samenumber of coordinates.
The Dot ProductDefinition
Example
Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then
#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)
= (2) + (18) + (−4)
= 16
Example
The dot product of
1−2
0
and
−1−2−1
0
is not defined!
The Dot ProductDefinition
Example
Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then
#»v · #»w =
〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)
= (2) + (18) + (−4)
= 16
Example
The dot product of
1−2
0
and
−1−2−1
0
is not defined!
The Dot ProductDefinition
Example
Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then
#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉=
(−1)(−2) + (−6)(−3) + (1)(−4)
= (2) + (18) + (−4)
= 16
Example
The dot product of
1−2
0
and
−1−2−1
0
is not defined!
The Dot ProductDefinition
Example
Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then
#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)
=
(2) + (18) + (−4)
= 16
Example
The dot product of
1−2
0
and
−1−2−1
0
is not defined!
The Dot ProductDefinition
Example
Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then
#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)
= (2) + (18) + (−4)
=
16
Example
The dot product of
1−2
0
and
−1−2−1
0
is not defined!
The Dot ProductDefinition
Example
Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then
#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)
= (2) + (18) + (−4)
= 16
Example
The dot product of
1−2
0
and
−1−2−1
0
is not defined!
The Dot ProductDefinition
Example
Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then
#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)
= (2) + (18) + (−4)
= 16
Example
The dot product of
1−2
0
and
−1−2−1
0
is not defined!
The Dot ProductProperties
Algebraic Properties of the Dot Product
The dot product obeys the following laws.
commutative #»v · #»w = #»w · #»v
distributive #»v · ( #»w + #»x ) = #»v · #»w + #»v · #»x
associative c · ( #»v · #»w ) = (c · #»v ) · #»w
The Dot ProductProperties
Algebraic Properties of the Dot Product
The dot product obeys the following laws.
commutative #»v · #»w = #»w · #»v
distributive #»v · ( #»w + #»x ) = #»v · #»w + #»v · #»x
associative c · ( #»v · #»w ) = (c · #»v ) · #»w
The Dot ProductProperties
Algebraic Properties of the Dot Product
The dot product obeys the following laws.
commutative #»v · #»w = #»w · #»v
distributive #»v · ( #»w + #»x ) = #»v · #»w + #»v · #»x
associative c · ( #»v · #»w ) = (c · #»v ) · #»w
The Dot ProductLengths
ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have
#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n
= ‖ #»v ‖2
The dot product can be used to compute lengths!
The Dot ProductLengths
ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have
#»v · #»v =
v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n
= ‖ #»v ‖2
The dot product can be used to compute lengths!
The Dot ProductLengths
ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have
#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn=
v21 + v22 + · · ·+ v2n
= ‖ #»v ‖2
The dot product can be used to compute lengths!
The Dot ProductLengths
ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have
#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n
=
‖ #»v ‖2
The dot product can be used to compute lengths!
The Dot ProductLengths
ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have
#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n
= ‖ #»v ‖2
The dot product can be used to compute lengths!
The Dot ProductLengths
ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have
#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n
= ‖ #»v ‖2
The dot product can be used to compute lengths!
The Dot ProductAngles
QuestionHow can we measure the angle θbetween two vectors #»v and #»w?
#»v
#»w
θ
#»v − #»w
AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.
law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ
dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
The Dot ProductAngles
QuestionHow can we measure the angle θbetween two vectors #»v and #»w?
#»v
#»w
θ
#»v − #»w
AnswerForm a triangle.
Measure ‖ #»v − #»w‖2 in two ways.
law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ
dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
The Dot ProductAngles
QuestionHow can we measure the angle θbetween two vectors #»v and #»w?
#»v
#»w
θ
#»v − #»w
AnswerForm a triangle.
Measure ‖ #»v − #»w‖2 in two ways.
law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ
dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
The Dot ProductAngles
QuestionHow can we measure the angle θbetween two vectors #»v and #»w?
#»v
#»w
θ
#»v − #»w
AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.
law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ
dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
The Dot ProductAngles
QuestionHow can we measure the angle θbetween two vectors #»v and #»w?
#»v
#»w
θ
#»v − #»w
AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.
law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ
dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
The Dot ProductAngles
QuestionHow can we measure the angle θbetween two vectors #»v and #»w?
#»v
#»w
θ
#»v − #»w
AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.
law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ
dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
The Dot ProductAngles
QuestionHow can we measure the angle θbetween two vectors #»v and #»w?
#»v
#»w
θ
#»v − #»w
AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.
law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ
dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
The Dot ProductAngles
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
Corollary (The Cauchy-Schwarz Inequality)
| #»v · #»w | ≤ ‖ #»v ‖ · ‖ #»w‖
Example
Let θ be the angle between #»v = 〈1, 2, 3〉 and #»w = 〈1, 1, 1〉.Compute cos θ.
cos θ =#»v · #»w
‖ #»v ‖ · ‖ #»w‖=
(1)(1) + (2)(1) + (3)(1)√12 + 22 + 32 ·
√12 + 12 + 12
=6√
14 ·√
3
The Dot ProductAngles
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
Corollary (The Cauchy-Schwarz Inequality)
| #»v · #»w | ≤ ‖ #»v ‖ · ‖ #»w‖
Example
Let θ be the angle between #»v = 〈1, 2, 3〉 and #»w = 〈1, 1, 1〉.Compute cos θ.
cos θ =#»v · #»w
‖ #»v ‖ · ‖ #»w‖=
(1)(1) + (2)(1) + (3)(1)√12 + 22 + 32 ·
√12 + 12 + 12
=6√
14 ·√
3
The Dot ProductAngles
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
Corollary (The Cauchy-Schwarz Inequality)
| #»v · #»w | ≤ ‖ #»v ‖ · ‖ #»w‖
Example
Let θ be the angle between #»v = 〈1, 2, 3〉 and #»w = 〈1, 1, 1〉.Compute cos θ.
cos θ =#»v · #»w
‖ #»v ‖ · ‖ #»w‖=
(1)(1) + (2)(1) + (3)(1)√12 + 22 + 32 ·
√12 + 12 + 12
=6√
14 ·√
3
The Dot ProductAngles
Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ
Corollary (The Cauchy-Schwarz Inequality)
| #»v · #»w | ≤ ‖ #»v ‖ · ‖ #»w‖
Example
Let θ be the angle between #»v = 〈1, 2, 3〉 and #»w = 〈1, 1, 1〉.Compute cos θ.
cos θ =#»v · #»w
‖ #»v ‖ · ‖ #»w‖=
(1)(1) + (2)(1) + (3)(1)√12 + 22 + 32 ·
√12 + 12 + 12
=6√
14 ·√
3
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then
#»v · #»w > 0 means θ is
acute (0 ≤ θ < π/2)
#»v · #»w < 0 means θ is
obtuse (π/2 < θ ≤ π)
#»v · #»w = 0 means θ is
right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal? Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then#»v · #»w > 0 means θ is
acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is
obtuse (π/2 < θ ≤ π)
#»v · #»w = 0 means θ is
right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal? Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)
#»v · #»w < 0 means θ is
obtuse (π/2 < θ ≤ π)
#»v · #»w = 0 means θ is
right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal? Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is
obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is
right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal? Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)
#»v · #»w = 0 means θ is
right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal? Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is
right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal? Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal? Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal? Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal?
Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
TheoremLet θ be the angle between two vectors #»v 6= #»
O and #»w 6= #»
O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is right (θ = π/2)
Definition#»v and #»w are orthogonal if #»v · #»w = 0
Example
Are #»v = 〈1,√
2, 1, 0〉 and #»w = 〈1,−√
2, 1, 1〉 orthogonal? Yes,since
#»v · #»w = (1)(1) + (√
2)(−√
2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w =
0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0.
This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 =
3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3.
The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
=
w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.”
The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .
The Dot ProductAngles
ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .
The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation
0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3
Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by
#»w =
w1
w2
w3
=
3w2 − 8w3
w2
w3
= w2
310
+ w3
−801
where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .