If a⃗ ,b⃗ ,c⃗ are three non-zero vectors and n is a unit vector perpendicular to c⃗ such that a⃗ = α b⃗ - n,(α ≠ 0) and b⃗ .c⃗ = 12, then | c⃗ × (a⃗ × b⃗ ) | is equal to :

Solution

Maths sir · 4 steps

Expand with the BAC minus CAB rule, use n · c⃗ = 0, and the vector collapses to -12n.

1. Expand the product. Use p⃗ × (q⃗ × r⃗) = (p⃗ · r⃗)q⃗ - (p⃗ · q⃗)r⃗ with p⃗ = c⃗, q⃗ = a⃗, r⃗ = b⃗.

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

2. Find the dot products. c⃗ · b⃗ = 12 is given. Since n is perpendicular to c⃗, c⃗ · a⃗ = α(b⃗ · c⃗) - n · c⃗ = 12α - 0 = 12α.
3. Substitute. The vector becomes 12a⃗ - 12αb⃗ = 12(a⃗ - αb⃗). From the given relation, a⃗ - αb⃗ = -n, so the vector is -12n.
4. So. Since n is a unit vector, | -12n | = 12. The answer is (d).

The idea to keep. Whenever you see a vector triple product, expand it with BAC minus CAB first. Then look for perpendicular vectors, because their dot products vanish and the expression usually simplifies.

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