Let a⃗, b⃗, c⃗ be three vectors such that
|a⃗|=√31, 4|b⃗|=|c⃗|=2 and 2(a⃗ × b⃗)=3(c⃗ × a⃗).
If the angle between b⃗ and c⃗ is (2 π)/3, then ((a⃗ × c⃗)/(a⃗ · b⃗))² is equal to __________.

Solution

Maths sir · 5 steps

Move everything to one side: a cross product equal to zero means the vectors are parallel.

1. Read the lengths. 4|b⃗| = |c⃗| = 2 gives |c⃗| = 2 and |b⃗| = 1/2. Then b⃗·c⃗ = 1/2 × 2 × cos 2π/3 = -1/2.
2. Parallel vectors. Since c⃗×a⃗ = -a⃗×c⃗, the condition becomes a⃗×(2b⃗ + 3c⃗) = 0⃗. So 2b⃗ + 3c⃗ = λa⃗ for some scalar λ.

   |2b⃗+3c⃗|² = 4|b⃗|² + 9|c⃗|² + 12 b⃗·c⃗ = 1 + 36 - 6 = 31

3. Find lambda. So 31λ² = 31, which gives λ = ± 1 and a⃗ = ±(2b⃗ + 3c⃗). The sign does not matter, because both parts of the answer get squared.
4. Both pieces. a⃗·b⃗ = ±(2 × 1/4 + 3 × (-1/2)) = ∓ 1. Since c⃗×c⃗ = 0⃗, a⃗×c⃗ = ± 2(b⃗×c⃗), with |b⃗×c⃗| = 1/2 × 2 × √3/2 = √3/2, so |a⃗×c⃗| = √3.
5. So. ((a⃗×c⃗)/(a⃗·b⃗))² = (|a⃗×c⃗|²)/(a⃗·b⃗)² = 3/1 = 3.

The common slip: 0.75. Dropping the factor 2 and writing a⃗×c⃗ = b⃗×c⃗ gives |a⃗×c⃗|² = 3/4, so the answer comes out as 0.75 instead of 3.

The idea to keep. When a⃗×p⃗ = a⃗×q⃗ in some form, rewrite it as a⃗×(p⃗-q⃗) = 0⃗. Then a⃗ is a multiple of p⃗-q⃗, and the given length fixes it.

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