Let a⃗ = î - α ĵ + β k̂, b⃗ = 3î + β ĵ - α k̂ and c⃗ = -α î - 2ĵ + k̂, where α and β are integers. If a⃗ . b⃗ = - 1 and b⃗ . c⃗ = 10, then ( a⃗ × b⃗ ). c⃗ is equal to ___________.

Solution

Maths sir · 4 steps

Use the two dot products to pin down α and β first. Then the triple product is just a 3 by 3 determinant.

1. Two equations. a⃗ · b⃗ = 3 - αβ - αβ = 3 - 2αβ = -1, so αβ = 2. Also b⃗ · c⃗ = -3α - 2β - α = -4α - 2β = 10, so 2α + β = -5.
2. Find α and β. Put β = -5 - 2α into αβ = 2 to get 2α² + 5α + 2 = 0. This gives α = -2 or α = -1/2. Since α must be an integer, α = -2 and β = -1.
3. Actual vectors. Now a⃗ = î + 2ĵ - k̂, b⃗ = 3î - ĵ + 2k̂ and c⃗ = 2î - 2ĵ + k̂. The value of (a⃗ × b⃗) · c⃗ is the determinant of their components.

   |1, 2, -1; 3, -1, 2; 2, -2, 1| = 1(-1+4) - 2(3-4) + (-1)(-6+2) = 3 + 2 + 4

4. So. (a⃗ × b⃗) · c⃗ = 9.

The common slip: 5. Forgetting the minus sign on the middle cofactor turns +2 into -2 and gives 5 instead of 9.

The idea to keep. When a vector question hides unknowns inside the components, use the given dot products to find them first. Let the integer condition choose the root. Then write the triple product as a determinant.

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