Let a⃗=α î+ĵ-k̂ and b⃗=2 î+ĵ-α k̂, α>0. If the projection of a⃗ × b⃗ on the vector -î+2 ĵ-2 k̂ is 30, then α is equal to :

Solution

Maths sir · 4 steps

Find the cross product in terms of α first, then the projection gives one quadratic in α.

1. Cross product. Expand the determinant with rows (α, 1, -1) and (2, 1, -α). This gives a⃗ × b⃗ = (1-α)î + (α² - 2)ĵ + (α - 2)k̂.
2. Projection formula. The projection of a vector on c⃗ = -î + 2ĵ - 2k̂ is its dot product with c⃗, divided by |c⃗| = √(1 + 4 + 4) = 3.

   (-(1-α) + 2(α² - 2) - 2(α - 2))/3 = (2α² - α - 1)/3 = 30

3. Solve the quadratic. 2α² - α - 91 = 0 factorises as (α - 7)(2α + 13) = 0. So α = 7 or α = -13/2.
4. So. Since α > 0, α = 7. Check: a⃗ × b⃗ = -6î + 47ĵ + 5k̂, and (6 + 94 - 10)/3 = 30. The answer is (d).

The idea to keep. Projection of u⃗ on v⃗ is (u⃗ · v⃗)/|v⃗|. Divide by the length of the vector you project onto, then use the given condition to drop the extra root.

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