Question

If `veca and vecb` are two non-collinear vectors, then find x, such that `vecα = (x - 2)veca + vecb and vecβ = (3 + 2x)veca - 2vecb` are collinear.

Answer 1

Given: `vecα = (x - 2)veca + vecb`

`vecβ = (3 + 2x)veca - 2vecb`

Since `veca and vecb` are non-collinear, they are linearly independent.

For `vecα and vecβ`​ to be collinear, their corresponding coefficients must be proportional.

∴ `(a_1)/(a_2) = (b_1)/(b_2)`

`(x - 2)/(3 + 2x) = 1/(-2)`

⇒ −2(x − 2) = 1(3 + 2x)

⇒ −2x + 4 = 3 + 2x

−2x − 2x = 3 − 4

−4x = −1

x = `(-1)/(-4)`

x = `1/4`