Vector problem?

Question

A and B are position vectors relative to and origin 'O'.

vector A is 4i + 14 j, and vector B is 12i + 2 j.

C lies on AB and position vector is 2t i + t j.

find the value of 'k' and the ratio of AC : CB.

thank you for your coorperation with me. the easiest way is going to get 10 points.

sorry find the value of 't'

Answer 1

I guess you need the value of "t", not of "k".

Points on AB satisfy the equation
[1] ... C = r A + (1-r) B
In your case,
[2] ... 2t i + t j = r (4 i + 14 j) + (1-r) (12 i + 2 j)
Splitting this into components,
[3a] ... 2t = 4r + 12(1-r) = 12 - 8r
[3b] ... t = 14r + 2(1-r) = 2 + 12r
Multiply second equation with 2,
[3c] ... 2t = 4 + 24r
enables us to eliminate t by subtraction,
[4] ... 0 = 8 - 32r

so that r = 1/4, and (1-r) = 3/4, showing that the ratio AC : CB = 3:1.

Next, we solve for t (using [3a] or [3b]):
[5a] ... 2t = 12 - 8r = 12 - 2 = 10 --> r = 5
[5b] ... t = 2 + 12r = 2 + 3 = 5

So C is point 10 i + 5 j.