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Nat 5
VectorsVectors
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Vectors and ScalarsAdding / Sub of vectorsPosition Vector
Exam Type Questions
3D Vectors
Vector Journeys
Magnitude of a Vector
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Nat 5
Vectors & Scalars
A vector is a quantity with
BOTH magnitude (length) and direction.
Examples : Gravity
Velocity
Force
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Vectors & Scalars
A scalar is a quantity that has
magnitude ONLY.
Examples : Time
Speed
Mass
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Vectors & Scalars
A vector is named using the letters at the end of the directed line segment
or
using a lowercase bold / underlined letter
This vector is named
AB))))))))))))))
AB))))))))))))))
or u
or u
uu
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Vectors & Scalars
A vector may also be
represented in component form.
4
2CD w
))))))))))))))
w
xAB d
y
))))))))))))))
z
2
1FE z
))))))))))))))
Also known as column vector
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Equal Vectors
Vectors are equal only if they both have
the same magnitude ( length ) and direction.
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Equal Vectors
Which vectors are equal.
a
b c
d
ef
g
h
a
g
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Equal Vectors
Sketch the vectors 2a , -b and 2a - b
a b
2a
-b
2a
-b
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Nat 5
19 Apr 202319 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try N5 TJ
Ex 15.1
Ch15 (page 143)
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VectorsVectors
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Addition of Vectors
Any two vectors can be added in this way
a
b
a + b
b
2
4
6
3
8
1
Arrows must be
nose to tail
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Addition of Vectors
Addition of vectors
3 2Let and
4 5AB BC
))))))))))))))))))))))))))))
A
B
C
Then + AB BC AC))))))))))))))))))))))))))))))))))))))))))
5So
1AC
))))))))))))))
3 2 5 +
4 5 1
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Nat 5
Addition of Vectors
In general we have
If and then a c
u vb d
For vectors u and v
+ + = a c a c
u vb d b d
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Zero Vector
The zero vector
1 If AB
2
))))))))))))))
1 1 0 AB + BA + =
2 2 0
))))))))))))))))))))))))))))
0 is called the zero vector, written 0
0
1 then BA
2
))))))))))))))
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Nat 5
Subtraction of Vectors
u
v
u + (-v) =
6
3
5
0
Notice arrows nose
to tail
1
3
-v
u - v
Subtracting vectors
think adding a negative vector
u + (-v)
1
3
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Subtraction of Vectors
Subtraction of vectors
6 2Let and
5 4a b
Then ( )a b a b
6 2 4 -
5 4 1
6 2 4
5 4 1
a b
a - b
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Nat 5
Subtraction of Vectors
In general we have
If and then a c
u vb d
For vectors u and v
= a c a c
u vb d b d
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Nat 5
19 Apr 202319 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try N5 TJ
Ex 15.2
Ch15 (page 145)
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VectorsVectors
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Position Vectors
A is the point (3,4) and B is the point (5,2).
Write down the components of
OA ))))))))))))))
OB OA ))))))))))))))))))))))))))))
3
4
B
A
OB )))))))))))))) 5
2
AB )))))))))))))) 2
2
5 3 2
2 4 2
Answers
the same !
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Position Vectors
is called the position vector of the point A
relative to the origin , written as
OA))))))))))))))
O a
is called the position vector of the point B
relative to the origin , written as
OB))))))))))))))
O b
AB ))))))))))))))
b a AO OB ))))))))))))))))))))))))))))
a b
B
A
a
b0
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Position Vectors
where and are the position vectors of and .
AB b a
a b A B
))))))))))))))
B
A
a
b0
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Nat 5
Position Vectors
If P and Q have coordinates (4,8) and (2,3)
respectively, find the components of
2 4
3 8
PQ))))))))))))))
2
5
PQ q p ))))))))))))))
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Nat 5
P
Q
O
Position Vectors
Graphically
P (4,8)
Q (2,3)
= PQ)))))))))))))) 2
5
p
q
q - p
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Nat 5
19 Apr 202319 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try N5 TJ
Ex 15.3
Ch15 (page 146)
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Position VectorsPosition Vectors
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Magnitude of a Vector
A vector’s magnitude (length) is represented by
or PQ u))))))))))))))
A vector’s magnitude is calculated using Pythagoras
Theorem.
2 2PQ u a b )))))))))))))) u
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Nat 5
Calculate the magnitude of the vector.
4
2CD w
))))))))))))))w
2 2w a b
2 24 2w
20w
4 5 2 5w
Magnitude of a Vector
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Nat 5
Calculate the magnitude of the vector.
4
3PQ
))))))))))))))
2 2w a b
2 2( 4) 3w
16 9w
5w
Magnitude of a Vector
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Nat 5
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Now try N5 TJ
Ex 15.4
Ch15 (page 147)
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Position VectorsPosition Vectors
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Nat 5
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Vector JourneysVector Journeys
As far as the vector is concerned, only the
FINISHING POINT
in relation to the
STARTING POINT
is important.
The route you take is IRRELEVANT.
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Nat 5
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Vector JourneysVector Journeys
W Xu
v
YZ
M
Given that WXZY twiceis find ZMXMWZWY and , ,
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Vector JourneysVector Journeys
W Xu
v
YZ
M
vu ZM2
12
2u
vXY XM2
1
2
1
uvu WZ 2
vu WY
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Nat 5
19 Apr 202319 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try N5 TJ
Ex 15.5
Ch15 (page 149)
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Vector JourneysVector Journeys
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O
3D Coordinates
In the real world points in space can be located using a 3D coordinate system.
x
y
z
For example, air traffic controllers find the location a plane by its height and grid reference.
(x, y, z)
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O
3D Coordinates
Write down the coordinates for the 7 vertices
x
yzA
B
C
D
E
F
G
What is the coordinates of the vertex H so that it makes a cuboid shape.
6
2
1
(6, 1, 2)
(6, 0, 2)
(6, 0, 0)
(6, 1, 0)
(0, 1, 2)
(0, 0, 2)
(0,0, 0)
H(0, 1, 0 )
H
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3D Vectors
Good NewsAll the rules for 2D vectors apply in the same way for
3D.
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Addition of Vectors
Addition of vectors
3 2
Let 4 and 5
1 2
u v
Then + u v
3 2 5
4 + 5 1
1 2 3
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Nat 5
Addition of Vectors
In general we have
If and then
a d
u b v e
c f
For vectors u and v
+ + =
a d a d
u v b e b e
c f c f
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Magnitude of a Vector
A vector’s magnitude (length) is represented by
v
A 3D vector’s magnitude is calculated using Pythagoras Theorem twice.
2 2 2v x y z
z v
2
3
1
O x
y2 2 23 2 1v
14v
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Subtraction of Vectors
Subtraction of vectors
6 2
Let 5 and 4
3 2
a b
Then a b
6 2 4
5 - 4 1
3 2 1
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Subtraction of Vectors
If and then
a d
u b v e
c f
For vectors u and v
=
a d a d
u v b e b e
c f c f
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Position Vectors
The position vector of a 3D point A is ,
usually written as
OA))))))))))))))
a
3
= = 2
1
OA
))))))))))))))a
z a
2
3
1
O x
y
A (3,2,1)
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Position Vectors
If R is (2,-5,1) and S is (4,1,-3) then RS s r ))))))))))))))
4 2 2
1 5 6
3 1 4
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Nat 5
19 Apr 202319 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try N5 TJ
Ex 15.6
Ch15 (page 150)
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3D Vectors3D Vectors
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Nat 5
Are you on Target !
• Update you log book
• Make sure you complete and correct
ALL of the Vector questions in the
past paper booklet.
