Question

Let ⃗α = (λ - 2) ⃗a + ⃗b and ⃗β = (4λ - 2) ⃗a + 3⃗b be two given vectors where ⃗a and ⃗b
are non collinear. The value of λ for which vectors ⃗α and ⃗β
are collinear, is : (A) –4 (B) –3
(C) 4 (D) 3

Answer 1

Answer:

(A) -4

Step-by-step explanation:

Since $\vec{\alpha}\ and\ \vec{\beta}$ are collinear, for some real number k,

$\alpha = k \ \beta => \alpha - k\ \beta = 0$

i.e, $((\lambda - 2) \vec{a} + \vec{b}) - k( (4\lambda - 2)\vec{a} + 3 \vec{b} ) = 0\\(\lambda - 2 - 4k\lambda + 2k)\vec{a} + (1 - 3k)\vec{b} = 0\\( (1-4k)\lambda -2(1-k))\vec{a} = (3k - 1)\vec{b}$

but, it is given that vectors a and b are non collinear. So, this means,

$(1-4k)\lambda - 2(1-k) = 0\\and\\3k - 1 = 0\\=> \lambda = \frac{2(1-k)}{1 - 4k} \ and\ k = \frac{1}{3}\\=> \lambda = -4$