Let a⃗=î+ĵ+k̂, b⃗=2 î+4 ĵ-5 k̂ and c⃗=x î+2 ĵ+3 k̂, x ∈ ℝ. If d⃗ is the unit vector in the direction of b⃗+c⃗ such that a⃗ · d⃗=1, then (a⃗ × b⃗) · c⃗ is equal to

Solution

Maths sir · 4 steps

First use a⃗ · d⃗ = 1 to find x, then the triple product is just a 3 by 3 determinant.

1. Set up d. b⃗+c⃗ = (x+2)î+6ĵ-2k̂. Since d⃗ is its unit vector, a⃗ · d⃗ = (a⃗ · (b⃗+c⃗))/(|b⃗+c⃗|) = (x+6)/√((x+2)²+40).
2. Find x. Setting this equal to 1 gives x+6 = √((x+2)²+40). Squaring, x²+12x+36 = x²+4x+44, so 8x = 8 and x = 1. Check: 1+6 = 7 and √(9+40) = 7, so it works.

   (a⃗ × b⃗) · c⃗ = |1, 1, 1; 2, 4, -5; 1, 2, 3|

3. Expand. Expanding along the first row gives 1(12+10) - 1(6+5) + 1(4-4) = 22 - 11 + 0 = 11.
4. So. (a⃗ × b⃗) · c⃗ = 11, which is option (c).

The idea to keep. A dot product with a unit vector is the dot product divided by the length. After you square to find the unknown, check the answer in the original equation. Then the triple product is a determinant.

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