Let u⃗ = î - ĵ - 2k̂,v⃗ = 2î + ĵ - k̂,v⃗ . w⃗ = 2 and v⃗ × w⃗ = u⃗ + λ v⃗ . Then u⃗ . w⃗ is equal to :

Solution

Maths sir · 4 steps

Dot both sides with v⃗, then with w⃗: a cross product is perpendicular to both.

1. Use perpendicularity. A cross product is perpendicular to both vectors in it. So v⃗ · (u⃗ + λ v⃗) = 0 and w⃗ · (u⃗ + λ v⃗) = 0.
2. Find lambda. u⃗ · v⃗ = 2 - 1 + 2 = 3 and |v⃗|² = 4 + 1 + 1 = 6. So 3 + 6λ = 0, which gives λ = -1/2.
3. Dot with w. Expand the second condition and use the given v⃗ · w⃗ = 2.

   u⃗ · w⃗ + λ (v⃗ · w⃗) = 0

4. So. u⃗ · w⃗ = -λ × 2 = -(-1/2) × 2 = 1, which is option (d).

The idea to keep. Whenever a cross product equals some combination of vectors, dot both sides with the vectors inside the cross product. The left side becomes zero and the unknown constant comes out.

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