If a⃗ = 2î + ĵ + 2k̂, then the value of
| î × ( a⃗ × î ) |² + | ĵ × ( a⃗ × ĵ ) |² + | k̂ × ( a⃗ × k̂ ) |² is equal to____

Solution

Maths sir · 4 steps

Expand each triple product: it removes the matching component of a⃗, so each term is just two squares.

1. Expand one term. Use p⃗ × (q⃗ × r⃗) = (p⃗ · r⃗)q⃗ - (p⃗ · q⃗)r⃗. With p⃗ = r⃗ = î and q⃗ = a⃗, this gives î × (a⃗ × î) = a⃗ - (î · a⃗)î.

   î × (a⃗ × î) = a⃗ - a₁ î = a₂ ĵ + a₃ k̂

2. All three terms. The three squared lengths are a₂² + a₃², a₁² + a₃² and a₁² + a₂². Each component appears twice, so the sum is 2(a₁² + a₂² + a₃²) = 2|a⃗|².
3. Put in numbers. |a⃗|² = 2² + 1² + 2² = 9, so the sum is 2 × 9 = 18.
4. So. The required value is 18.

The common slip: 27. Taking each triple product as a⃗ itself, without removing the component along the unit vector, gives 3 × 9 = 27 instead of 18.

The idea to keep. n × (a⃗ × n) is the part of a⃗ perpendicular to n. Summed over î, ĵ, k̂, the squared lengths always give 2|a⃗|².

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