Let a⃗=3 î+ĵ and b⃗=î+2 ĵ+k̂. Let c⃗ be a vector satisfying a⃗ ×(b⃗ × c⃗)=b⃗+λ c⃗. If b⃗ and c⃗ are non-parallel, then the value of λ is :

Solution

Maths sir · 4 steps

Expand the triple product, then match the coefficients of b and c on both sides.

1. Expand. Use the BAC minus CAB rule on the left side.

   a⃗ × (b⃗ × c⃗) = (a⃗·c⃗) b⃗ - (a⃗·b⃗) c⃗ = b⃗ + λc⃗

2. Compare coefficients. Since b⃗ and c⃗ are non-parallel, they are linearly independent, so the coefficients on each side must match. This gives a⃗·c⃗ = 1 and λ = -(a⃗·b⃗).
3. Find the dot product. a⃗·b⃗ = (3)(1) + (1)(2) + (0)(1) = 5.
4. So. λ = -5, which is option (a).

The idea to keep. When a triple product equals a combination of two non-parallel vectors, expand it as (a⃗·c⃗)b⃗ - (a⃗·b⃗)c⃗ and compare coefficients. Watch the minus sign.

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