JEE Main 2023 · 24 January · evening · Single correct · +4 / −1
If two vectors P⃗=î+2mĵ+mk̂ and Q⃗=4î-2ĵ+mk̂ are perpendicular to each other. Then, the value of m will be :
- (a) -1
- (b) 3
- (c) 1
- (d) 2
Answer and solution
Answer (d) 2
Solution
Physics sir · 4 steps
Perpendicular means the dot product is zero, so multiply matching components and add them.
- Condition to use. Two vectors are perpendicular when P⃗ · Q⃗ = 0, because cos 90° = 0. The cross product is the test for parallel vectors, not perpendicular ones.
- Multiply components. Multiply the î, ĵ and k̂ parts pairwise and add them: (1)(4) + (2m)(-2) + (m)(m).
P⃗ · Q⃗ = 4 - 4m + m² = (m-2)² = 0
- Check the options. The other options do not give zero: m = -1 gives 9, m = 3 gives 1, and m = 1 gives 1.
- So. The quadratic is a perfect square, so its only root is m = 2. The answer is (d).
The idea to keep. Perpendicular means the dot product is zero. Parallel means the cross product is zero. Always include every component, including the k̂ term.
Answer from the NTA final answer key. Solution by PrepWiser.