The document discusses applying ratios to vectors. It provides examples of using ratios to find unknown points along vectors. The key steps are: 1) Form an equation using the position vector rule relating the unknown point P to the known endpoints A and B; 2) Make P the subject of the equation; 3) Solve the equation to find the coordinates of P. Examples show setting up equations for points dividing vectors in given ratios and solving to find the coordinates of the unknown points.

1. Block 3
Ratios

2. What is to be learned?
• How to apply ratios to vectors
(without panicking)
• Some parts of the bible may be open to
interpretation

3. yellow : red = 3:2

4. Lines
AB:GH = 4:1
AB = GH4

5. David: Goliath = 4:1
Length of David = 4 X Length of Goliath
David Goliath

6. Applying to Vectors
A (2,5,3) B (5,8,9)
P divides AB in ratio 2:1
A BP2 1
AP = PB2

7. Applying to Vectors
A (2,5,3) B (5,8,9)
AP = PB2
use position
vector rule
p – a = 2(b – p)
p – a = 2b – 2p
3p = 2b + a
=2 +
( )
=
( )12
21
21
3p
p =
( )4
7
7P = (4,7,7)
( ) 2
5
3
+
5
8
9
→ →

8. position vectorsp – a = 2(b – p)
p – a = 2b – 2p
( )
3p = 2b + a
= +( )9
2
8
=
( )
21
0
18
3p p =
( )
7
0
6P = (7,0,6)
3
-4
2
AP = PB2
→ →
A BP2 1
A (3,-4,2) B (9,2,8)
P divides AB in ratio 2:1
2

9. Vectors, Ratios
and Unknown Points
• Form a wee equation
(diagram v helpful)
• Use position vector rule
• Make required vector the subject
• Remember to change back to coordinates!

10. A (2,5,3) B (6,5,7)
P divides AB in ratio 1:3
A BP1 3
AP = PB3
→ →

11. position vectors3(p – a) = b – p
3p – 3a = b – p
( )
4p = b + 3a
= +3( )6
5
7
=
( )
12
20
16
4p p =
( )3
5
4
P = (3,5,4)
2
5
3
AP = PB3
→ →

12. position vectors3(p – a) = 2(b – p)
3p – 3a = 2b – 2p
( )
5p = 2b + 3a
= +3( )68
9
=
( )
15
25
30
5p p =
( )
3
5
6P = (3,5,6)
1
3
4
AP = PB2
→ →
A BP2
A (1,3,4) B (6,8,9)
P divides AB in ratio 2:3
2
3
3
Key
Question

13. External ratios
P divides AB in ratio 2:1externally
A B P
2
1
AP = BP2
→ →
AP= PB-2
→ →
Or