example 1 if p is the point (5, 1) and q is the point (7, 3). find
TRANSCRIPT
Example 1
If P is the point (5, 1) and Q is the point (7, 3). Find PQ��������������
PQ q p ��������������
7 5
3 1
2
4
Example 2
If A is the point (3,-1, 9) and B is the point (8, -1, 5).
Find AB��������������
AB b a ��������������
8 3
1 1
5 9
5
0
4
Example 3
For the points M(-2, 1, 5)) and N(-3, -5, 6), find the
components of and calculate its magnitude.MN��������������
MN n m ��������������
3 2
5 1
6 5
1
6
1
Find Components
1
6
1
MN
��������������
2 2 2( 1) ( 6) 1MN ��������������
1 36 1
38
Calculate magnitude
Example 4a(i).
Find s + t - q
9
6s
2
3q
s t q
9 4·5 2
6 3 3
11·5
0
Find Components
4·5
3t
Example 4a(ii).
Find | r + u |
6
4r
9
6u
r u
6 9
4 6
15
2
Find Components
15
2r u
2 2( 15) ( 2)r u
225 4
229
Calculate magnitude
Example 4b.
Which of the given vectors are parallel to v
3
2v
3
2p
For parallel vectors a = k b where k is a non-zero constant
2
3q
6
4r
9
6s
4·5
3t
9
6u
⇒
⇒
⇒
⇒
⇒
⇒
1p v
q vk
2r v
3s v
u vk
⇒ Parallel
⇒ Non-parallel
⇒ Parallel
⇒ Parallel
⇒ Non-parallel
⇒ Non-parallel
t vk
Example 5
If A is the point (3, 4) and B is the point (5, 2).
(i) Write down a and b
(ii) Find AB��������������
AB b a ��������������
3
4a
5 3
2 4
5
2b
2
2
Example 6
If L is the point (-6, 3, 1) and M is the point(8, 7, 2),
find . LM��������������
LM m l ��������������
6
3
1
l
8
7
2
m
6 8
3 7
1 2
14
4
1
Example 7
Prove that P (3, 4, 1), Q(9, 1, -5) and R(11, 0, -7) are collinear.
PQ q p ��������������
For vectors to be collinear they must be parallel and have a common point.
P kQ QR����������������������������
9 3
1 4
5 1
6
3
6
QR r q ��������������
11 9
0 1
7 5
2
1
2
× 3
× 3
× 3
so 3Q QP R����������������������������
The vectors are parallel and have the point Q in common
so they are collinear
Example 8
A is the point (3, -2, 4) and B is (3, 4, -1).
Find the coordinates of the point P which divides AB internally in the ratio 1:2
A B
m:n(3, -2, 4) (3, 4, -1)
1 : 2
1
1
2
2 3 3xQ
9
33
6 3
3
12
2
2 4
1YQ
00
3
4 4
3
1 1
2
2 4
1zQ
7
3
8 1
3
733,0,Q
Example 9
Find the midpoint of the line joining C(-2, 3, 1) and D(-8, -7, 1),.
1
2m c d
2
3
1
c
8
7
1
d
2 81
3 72
1 1
101
42
2
Midpoint M is (-5, 2, 1)
5
2
1
Example #
Under Construction
Further
Examples