Block 3

Ratios

What is to be learned?

• How to apply ratios to vectors(without panicking)

• Some parts of the bible may be open to interpretation

yellow : red = 3:2

Lines

AB:GH = 4:1

AB = GH4

David: Goliath = 4:1

Length of David = 4 X Length of Goliath

David Goliath

Applying to VectorsA (2,5,3) B (5,8,9)P divides AB in ratio 2:1

A BP2 1

AP = PB2

Applying to VectorsA (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

2121

3pp =( )4

7 7P = (4,7,7)

( ) 253

+589

→ →

position vectors p – 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

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!

A (2,5,3) B (6,5,7)P divides AB in ratio 1:3

A BP1 3

AP = PB3→ →

position vectors 3(p – a) = b – p 3p – 3a = b – p

( ) 4p = b + 3a

= +3( )657

= ( )12 2016

4p p =( )3 5 4

P = (3,5,4)

253

AP = PB3→ →

position vectors 3(p – a) = 2(b – p) 3p – 3a = 2b – 2p

( ) 5p = 2b + 3a

= +3( )689

= ( )152530

5p p =( )3 5 6P = (3,5,6)

134

AP = PB2→ →

A BP2

A (1,3,4) B (6,8,9)P divides AB in ratio 2:3

2

3

3

Key Question

External ratios

P divides AB in ratio 2:1externally

A B P

2

1AP = BP2→ →

AP= PB-2→ →Or