VectorsVectors are represented by a directed line segment its length representing the magnitude and an arrow indicating the direction
A
B
orAB��������������
u
u
This vector is named
CD��������������
v= =6
2
DC��������������
= -v =6
2
EF��������������
w ==4
3
FE��������������
= -w =4
3
These are also known as COLUMN VECTORS
C
Dv
E
F
w
C
Dv
E
F
w
CD��������������
v= =6
2
This is also known as the components of v
Q
Pa
b
It can be calculated using Pythagoras
If a
PQb
��������������
The magnitude of a vector is represented by its
length and is written or AB s��������������
2 2PQ a b ��������������
Example If7
find s1
s
s 2 2a b
2 27 (1) 50 5 2
Vectors are only equal if they have the samemagnitude and direction.
Equal Vectors
a
bu
cd
v
If u v then a c and b d
au
b
For vectors
andc
vd
cv
d
For vectors and
ew
f
Addition Of Vectors
then the resultant vector
v w c e
d f
v w
v w
AB��������������
=5
4
BC��������������
=3
1
AC��������������
i) Find the components of
AC ��������������
5 3
4 1
AB��������������
BC��������������
=
5 3
4 ( 1)
=8
3
The Zero Vector0
0
is called the zero vector written 0
0
0
AB��������������
BA��������������
5
3
5
3
If5
3
AB��������������
find the components of
AB��������������
BA��������������
Subtraction of Vectors
For vectors and if and a c
u v u vb d
a cu v
b d
Page 236 Exercise 13D questions 1 to 3
Multiplication by a scalar
If then and vector is parallel to a ka
u ku ku ub kb
u
ku
Hence if then u is parallel to u kv v
Conversely if is parallel to then uu v kv
Unit Vectors
For any vector v there exists a parallel vector u of magnitude 1 unit. This is called a Unit Vector.
i.e. Find the components of the unit vector u parallel to vector3
4v
2 23 4v
5
Since the magnitude of v is 5, the unit vector u must be 1/5 v
1
5u v
31
45
35
45
Position Vectors
where and are the position vectors of A and B.AB b a a b ��������������
If P and Q have coordinates (4,8) and (2,3) find the components of PQ��������������
PQ q p ��������������
2 4
3 8
2
5
Collinearity
We have seen that if a vector v = ku then v must be parallel to u.
If vectors v and u also have a point in common then because they are parallel they must lie on the same line so by definition must be collinear.
If where is a scalar then is parallel to BC.
If B is also a point in common to both AB and BC, then
A, B and C are said to be collinear.
AB k BC k AB��������������������������������������������������������
Prove that the points A(2,4), B(8,6) and C(11,7) are collinear.
AB b a ��������������
8 2
6 4
6
2
BC c b ��������������
11 8
7 6
3
1
2 AB BC����������������������������
is parallel to AB BC����������������������������
B is a point in common to both AB and BC so A, B and C are collinear
Section Formula
If p is the position vector of the point P that divides AB in the ratio m:n then:
n mp a b
m n m n
A
B
Pm
n
A and B have coordinates (3,2) and (7,14) respectively. Find the coordinates of the point P that divides AB in the ratio 1:3
1. Draw a quick sketch
(3,2)
(7,14)
P1
3
3 73 1
2 144 4p
3 713
2 144
9 71
6 144
4
5
the coordinates of P are (4,5)
3 Dimensional Vectors
x
y
z
A
The point A has a position relative to the x y and z axis
3
4
6
A(3, 4, 6)
Find the coordinates of P
y
x
z
P
4
2
1
P(4, 2, 1)
O
Find the coordinates of Q
y
x
z
Q
-1-2
-3
Q(-1, -2, -3)
3D Unit VectorsA vector can also be defined in terms of i, j and k where i, j and k are unit vectors in the x, y, and z directions respectively.
y
x
z
i 1
j 11 k
1
0
0
i
0
1
0
j
0
0
1
k
In component form the vectors are written as
Any vector can be expressed as a combination of its components.
3
4 can be written
2
v
3 4 2v i j k
Properties of 3D vectors
P
Q
42
5
4
2
5
PQ
��������������
2 2 2PQ PR QR
R
S2 2 2PS SR QR 2 2 24 2 5PQ
45
3 5
2 2 2If then
a
PQ b PQ a b c
c
����������������������������
2 2If then
a
PQ b PQ a b c
c
����������������������������
If , then
a d a d
u b v e u v b e
c f c f
Addition / Subtraction
If then k
a ka
u b u kb
c kc
Scalar
If (a, b, c) and (d, e, f) then R S RS s r ��������������
Position Vector
Section Formula
If A is (4, -6, 12) and B is (4, 4, -3). Find P that divides AB in the ratio 3:2��������������
A (4, -6, 12)
B (4, 4, -3)
P3
2
4 42 3
6 45 5
12 3
p
8 121
12 125
24 9
20 41
0 05
15 3
(4,0,3)p
The Scalar Product
a
b
For two vectors a and b, the scalar product is defined by
. cosa b a b
Where is the angle between a and b, 00 180
The scalar product is also known as the dot product.
The vectors must be directed away from the point of intersection.
0 0
Find the scalar product for vectors and when
4 units and 5 units for
(a) 45 ( ) 90
a b
a b
b
( ) . cosa a b a b 04 5 cos 45
1220
10 2
( ) . cosb a b a b 04 5 cos90
20 0
0
If a and b are perpendicular then a . b = 0
Component Form of a Scalar Product
1 1
2 2
3 3
If and
a b
a a b b
a b
1 1 2 2 3 3then .a b a b a b a b
1 2
Find . for vectors 3 , 6
4 2
u w u w
. 1 2 3 6 ( 4) 2u w
12
Angle between vectors
We have seen that . cosa b a b
1 1 2 2 3 3and that .a b a b a b a b
1 1 2 2 3 3.Hence cos =
a b a b a ba b
a b a b
NOTE: if 0, 0 and . 0a b a b 0cos 0 90
Hence if . 0, and are perpendiculara b a b
Calculate the angle between vectors
3 +2j 5 and 4 j 3p i k q i k
9 4 25 38p 16 1 9 26q
3 4 2 1 5 3cos
38 26
0.9230 022.7 0 180
Other Vector FactsFor vectors and , . .a b a b b a
PROOF
1 1 2 2 3 3.a b a b a b a b
1 1 2 2 3 3b a b a b a
.b a
For vectors , and , . . .a b c a b c a b a c
PROOF
1 1 2 2 3 31 2 3.a b c a b c a b c a b c
1 1 1 1 2 2 2 2 3 3 3 3a b a c a b a c a b a c
1 1 2 2 3 3 1 1 2 2 3 3a b a b a b a c a c a c
. .a b a c
Calculate . when
3, 3, 4
p r q
p r q
300
300
p
r
q
. . .p r q p r p q
0 03 3cos30 3 4cos60
9 36
2
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