Let a⃗, b⃗ and c⃗ be three non-zero vectors such that b⃗ and c⃗ are non-collinear. If a⃗+5 b⃗ is collinear with c⃗, b⃗+6 c⃗ is collinear with a⃗ and a⃗+α b⃗+β c⃗=0⃗, then α+β is equal to

Solution

Maths sir · 4 steps

Write both collinearity facts as equations, then kill the coefficients of b⃗ and c⃗.

1. Translate collinearity. Collinear means one vector is a scalar multiple of the other. So a⃗+5b⃗=λc⃗ and b⃗+6c⃗=μa⃗ for some scalars λ and μ.
2. Eliminate a. From the first equation, a⃗=λc⃗-5b⃗. Put this into the second equation and collect terms.

   (1+5μ) b⃗+(6-λμ) c⃗=0⃗

3. Use non-collinearity. Since b⃗ and c⃗ are non-collinear, both coefficients must be zero. So μ=-1/5 and λμ=6, which gives λ=-30.
4. So. The first equation becomes a⃗+5b⃗=-30c⃗, that is a⃗+5b⃗+30c⃗=0⃗. Hence α=5, β=30 and α+β= 35, option (d).

The idea to keep. If pb⃗+qc⃗=0⃗ with b⃗, c⃗ non-collinear, then p=q=0. Turn every collinearity condition into a scalar multiple, eliminate one vector, and compare coefficients.

Answer from the NTA final answer key. Solution by PrepWiser.