Question

Relative to an origin O, the position vectors of points A, B and C are -5i-11j, 23i-4j and λ(i-3j) respectively. Given that C lies on the line AB, find the value of λ

Answer 1

λ = 3

Explanation:

You want the value of λ that places point C = λ(i -3j) on the line between A(-5i -11j) and B(23i -4j).

Solution

The parameterized line between A and B is ...

P = A +t(B -A)

where P is a point on AB. We can set P = C to find the value of t that identifies the point on AB. Comparing that to C, we can determine the value of λ.

λ(i -3j) = -5i -11j +t(23i -4j -(-5i -11j)) = (-5+28t)i +(-11 +7t)j

Equating components gives two equations in the unknowns λ and t:

• λ = -5 +28t
• -3λ = -11 +7t

Multiplying the second equation by -4 and adding the first gives ...

-4(-3λ) +λ = -4(-11 +7t) +(-5 +28t)

13λ = 39 . . . . . . . simplify

λ = 3 . . . . . . . . . divide by 13