Let the volume of a parallelopiped whose coterminous edges are given by
u⃗ = î + ĵ + λ k̂, v⃗ = î + ĵ + 3k̂ and
w⃗ = 2î + ĵ + k̂ be 1 cu. unit. If θ be the angle between the edges u⃗ and w⃗ , then cosθ can be :

Solution

Maths sir · 3 steps

Volume gives you λ through the triple product, then a plain dot product gives you cos θ.

1. Volume as determinant. The volume of the parallelepiped is |[u⃗ v⃗ w⃗]|, the absolute value of the determinant of the three edge vectors.

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

2. Find lambda. The volume is 1, so |3 - λ| = 1. This gives λ = 2 or λ = 4, and both must be checked.

   λ	u⃗ · w⃗ = 3 + λ	|u⃗| |w⃗| = √(2+λ²) √6	cos θ
   2	5	√6×√6 = 6	5/6, not an option
   4	7	√18×√6 = √108 = 6√3	7/6√3

3. So. With λ = 4, cos θ = 7/6√3, which is option (a).

The idea to keep. Volume of a parallelepiped is the absolute value of the triple product, so setting it equal to a number gives two cases. Always try both before ruling out options.

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