Let a⃗ = î + ĵ + 2k̂ and b⃗ = - î + 2ĵ + 3k̂. Then the vector product ( a⃗ + b⃗ ) × ( ( a⃗ × ( ( a⃗ - b⃗ ) × b⃗ ) ) × b⃗ ) is equal to :

Solution

Maths sir · 5 steps

Use b⃗ × b⃗ = 0⃗ twice and the whole chain shrinks to one cross product.

1. Innermost bracket. (a⃗ - b⃗) × b⃗ = a⃗ × b⃗ - b⃗ × b⃗ = a⃗ × b⃗, because any vector crossed with itself is zero.
2. Expand and cross. By the triple product rule, a⃗ × (a⃗ × b⃗) = (a⃗ · b⃗)a⃗ - (a⃗ · a⃗)b⃗. Crossing this with b⃗ removes the b⃗ term, since b⃗ × b⃗ = 0⃗. That leaves (a⃗ · b⃗)(a⃗ × b⃗).

   Expression = (a⃗ · b⃗) [(a⃗ + b⃗) × (a⃗ × b⃗)]

3. Find the pieces. a⃗ · b⃗ = -1 + 2 + 6 = 7. a⃗ × b⃗ = (1· 3 - 2· 2)î - (1· 3 - 2·(-1))ĵ + (1· 2 - 1·(-1))k̂ = -î - 5ĵ + 3k̂. And a⃗ + b⃗ = 3ĵ + 5k̂.
4. Last cross product. (3ĵ + 5k̂) × (-î - 5ĵ + 3k̂) has components 3· 3 - 5·(-5) = 34, -(0· 3 - 5·(-1)) = -5 and 0·(-5) - 3·(-1) = 3. So it is 34î - 5ĵ + 3k̂.
5. So. The product is 7(34î - 5ĵ + 3k̂), which is option (b).

The idea to keep. In a long chain of cross products, hunt for a vector crossed with itself first. Simplify with b⃗ × b⃗ = 0⃗ and the triple product rule, then plug in numbers once.

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