1 Kidoguchi, Kenneth
§8.4 Vectors
Objectives
xi
yj
zk
0rrt < 0
t > 0
1. Graph Vectors
2. Find a Position Vector
3. Add & Subtract Vectors
Algebraically
4. Find a Scalar Multiple
and the Magnitude of a
Vector
5. Find a Unit Vector
6. Find a Vector from Its
Direction and
Magnitude
7. Model with Vectors
§8.4 Vectors
Vectors and Vector Notations
The term vector is used by scientists
to indicate a quantity (such as
displacement or velocity or force) that
has both magnitude and direction.
A vector is often represented by an
arrow. The length of the arrow
represents the magnitude of the vector
and the arrow points in the direction of
the vector.
A vector must be denoted by printing
a letter in boldface (v) or by putting an
arrow above the letter റ𝑣 . A symbol
that is not boldface or has no arrow
above it is a scalar and NOT a vector.
A scalar is NEVER equal to a vector!
A
B
𝐯 = 𝐴𝐵
21 November 2017 2 Kidoguchi, Kenneth
𝐯 = v = റ𝑣 ≠ 𝑣
v is the vector that is
directed from A to B
§8.4 Vectors
Vector Geometry
If the vector 𝐮 = 𝐶𝐷 has the same
length (aka magnitude) and the
same direction as 𝐯 even though it
is in a different position we say that
𝐮 and 𝐯 are equivalent (or equal)
and we write 𝐮 = 𝐯.
The vector −𝐯 = 𝐵𝐴 has the same
length as 𝐯 but points in the
opposite direction.
The zero vector, denoted by 𝟎, has
length 0. It is the only vector with
no specific direction.
C
D
𝐮 = 𝐶𝐷
A
B−𝐯 = 𝐵𝐴
A
B
𝐯 = 𝐴𝐵
21 November 2017 3 Kidoguchi, Kenneth
§8.4 Vectors
Vector Addition
A
B
𝐯 = 𝐴𝐵Definition of Vector Addition:
If 𝐯 and 𝐰 are vectors positioned
so the initial point (tail) of 𝐰 is at
the terminal point (arrow head)
of 𝐯, then the sum 𝐯 + 𝐰 is the
vector from the initial point (tail)
of 𝐯 to the terminal point (head)
of 𝐰.
C
𝐰 = 𝐵𝐶
𝐯 +𝐰 = 𝐴𝐶
21 November 2017 4 Kidoguchi, Kenneth
§8.4 Vectors
Basis Vectors and Vector Notation
A
B
𝐯 = 𝐴𝐵
C
𝐰 = 𝐵𝐶
i
j
Basis vectors are unit vectors
(i.e. vectors of unit magnitude)
and specify direction. We will
use Ƹ𝑖 (pronounced i cap) a unit
vector along the positive x-axis
and Ƹ𝑗 a unit vector along the
positive y-axis .
ˆ ˆ2
2, 1
2, 1
2
1
2,1
i j
v
ˆ ˆ2
1, 2
1, 2
1
2
1, 2
i j
w
21 November 2017 5 Kidoguchi, Kenneth
Verboten !!!
§8.4 Vectors
Basis Vectors and Vector Notation
A
B
𝐯 = 𝐴𝐵
C
𝐰 = 𝐵𝐶
𝐯 +𝐰 = AC
i
j
The vector sum of 𝐯 and 𝐰 is the
sum of their components.
2 1
1 2
3
1
v w
21 November 2017 6 Kidoguchi, Kenneth
§8.4 Vectors
Vector Magnitude and Trigonometric Form
A
B
𝐯 = 𝐴𝐵
i
j
q
cos
sin
cos
sin
v
v
v
q
q
q
q
v
2 2
For
tan
a
b
v
a b
b
a
q
v
v v
2 2
2
2
2 2
2tan
2 4
v
q q
v
v v
2 2 cos / 4
2 2 sin / 4
cos / 42 2
sin / 4
v
21 November 2017 7 Kidoguchi, Kenneth
§8.4 Vectors
Graphing Vectors Geometrically
21 November, 2017 8 Kidoguchi, Kenneth
i
j
w
c) 2wh
b) vw
Use the vectors illustrated to
graph the following vectors.
a) vwv
h
m112_vectorAddition.mw.
21 November, 2017 9 Kidoguchi, Kenneth
§8.4 Vectors
Graphing Vectors Geometrically
vw
vw
wv
i
j
h
Use the vectors illustrated to
graph the following vectors.
vw
2wh
2w
v
c) 2wh
b) vw
a) vw
m112_vectorAddition.mw.
§8.4 Vectors
Position Vectors
10 Kidoguchi, Kenneth
Suppose that 𝐯 is a vector with initial point P1 = (x1, y1), not necessarily the
origin, and terminal point P2 = (x2, y2). If 𝐯 = 𝑃1𝑃2 , then 𝐯 is equal to the
position vector:
2 1 2 1
2 1 2 1
2 1
2 1
,
ˆˆ
x x y y
i x x k y y
x x
y y
v
§8.4 Vectors
Resolving a Vector into Rectangular Components
21 November, 2017 11 Kidoguchi, Kenneth
x
y
i
j
1 2 3-3 -2 -1 0
1
2
3
-1
-2
-3
= (-2, 3)
= (3, -1)
(x2, y2)
(x1, y1)
v
Alternate Notation:
2
2
1
1
xv
y
x
y
2 1pv p
2p
1p
21 November, 2017 12 Kidoguchi, Kenneth
§8.4 Vectors
Resolving a Vector into Rectangular Components
x
y
i
j
1 2 3-3 -2 -1 0
1
2
3
-1
-2
-3
= (-2, 3)
= (3, -1)
,
5
2 3 1
4
3
,
(x2, y2)
(x1, y1)
v
Alternate Notation:
2
2
1
1
xv
y
x
y
2 3
3 1
5
4
2 1pv p
1 12 2,xx y y
2p
1p
§8.4 Vectors
Vector Arithmetic
13 Kidoguchi, Kenneth
Let 𝐯 = 𝑎1 , 𝑏1 and 𝐰 = 𝑎2 , 𝑏2 be two vectors and let a be a
scalar. Then:
1 2 1 2
1 2 1 2
1 2 1 2
1 2 1 2
1 1
1 1
2 2
1 1
a a a a
b b b b
a a a a
b b b b
a a
b b
v a b
v w
v w
v
v v
§8.4 Vectors
A Unit Vector in the Direction of 𝐯
For any nonzero vector 𝐯, the vector
is a unit vector of unit magnitude and has the same direction as
vector 𝐯. Hence:
v
ˆ
v
v
v
u
v
v
v
𝐯
ෝ𝐮
ˆ ˆ ˆ v u u v uv
21 November, 2017 15 Kidoguchi, Kenneth
§8.4 Vectors
Resolving a Vector into Rectangular Components
vw
vw 2ˆ ˆ2 2,1
1i j
)a v
) 2d wh
wv
i
j )e vw
h)c h
Sketch the given vectors and
express them in component
form.
)b w1
ˆ ˆ 1, 11
i j
0ˆ2 0,2
2j
0 1 22
2 1 0
1
2
vw
)f vw
2wh
2w
w
v
m112_vectorAddition.mw.
21 November, 2017 16 Kidoguchi, Kenneth
When the river is still, a boat
crosses it with velocity vector
റ𝑣𝑠ℎ𝑖𝑝 = 4,0 km/hr. Today, the
river flows with a velocity
റ𝑣𝑟𝑖𝑣𝑒𝑟 = 0,−3 km/hr. If the
river is 20 km wide, present the
analysis to find the:
§8.4 Vectors
Example Vector Application
a) resultant velocity vector of the boat.
b) resultant speed of the boat.
c) time it will take the boat to get from one river bank to the other if the
river is 20 km wide.
d) total distance travelled by the boat.
20 kmRiv
er B
ank
1
Riv
er B
ank
2
vrivervship
ij
§8.4 Vectors
Example Vector Application: An Object in Static Equilibrium
18 Kidoguchi, Kenneth
A box of supplies that weighs
1000 pounds is suspended by
two cables attached to the
ceiling as shown. Present the
analysis to find the tensions in
the two cables.
§8.4 Vectors
Example Vector Application: True Aircraft Speed & Direction
19 Kidoguchi, Kenneth
A Boeing 737 aircraft maintains a constant airspeed
of 600 miles per hour headed due south. The jet
stream is 60 miles per hour in the northeasterly
direction.
a) Express 𝑣𝑎 the velocity 737 the relative to 𝑣𝑤the velocity of the jet stream in terms if Ƹ𝑖 and Ƹ𝑗.
b) Find റ𝑣 the velocity of the 737 relative to the
ground.
c) Find the actual speed and direction of the 737
relative to the ground.