Let a⃗=2 î-ĵ+5 k̂ and b⃗=α î+β ĵ+2 k̂ . If ((a⃗ × b⃗) × î) · k̂=23/2 , then |b⃗ × 2 ĵ| is equal to :

Solution

Maths sir · 4 steps

Expand the triple product into 2b⃗ - αa⃗, and its k̂-part alone gives α.

1. Expand the product. Use (a⃗ × b⃗) × c⃗ = (a⃗ · c⃗)b⃗ - (b⃗ · c⃗)a⃗ with c⃗ = î. Here a⃗ · î = 2 and b⃗ · î = α.

   (a⃗ × b⃗) × î = 2b⃗ - αa⃗

2. Find alpha. Dotting with k̂ keeps only the k̂-components: 2(2) - α(5) = 4 - 5α = 23/2. So 5α = -15/2, which gives α = -3/2.
3. Cross with j. b⃗ × ĵ = α(î × ĵ) + 2(k̂ × ĵ) = αk̂ - 2î, and the βĵ term gives zero. Its magnitude is √(α² + 4) = √(9/4 + 4) = 5/2.
4. So. |b⃗ × 2ĵ| = 2 × 5/2 = 5, which is option (b). You never needed β.

The idea to keep. Do not compute a⃗ × b⃗ first. Expand (a⃗ × b⃗) × c⃗ = (a⃗ · c⃗)b⃗ - (b⃗ · c⃗)a⃗ and read off only the component asked for.

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