JEE Main 2023 · 1 February · evening · Single correct · +4 / −1
Let a⃗=5 î-ĵ-3 k̂ and b⃗=î+3 ĵ+5 k̂ be two vectors. Then which one of the following statements is TRUE ?
- (a) Projection of a⃗ on b⃗ is -13/√35 and the direction of the projection vector is opposite to the direction of b⃗.
- (b) Projection of a⃗ on b⃗ is 13/√35 and the direction of the projection vector is opposite to the direction of b⃗.
- (c) Projection of a⃗ on b⃗ is 13/√35 and the direction of the projection vector is same as of b⃗.
- (d) Projection of a⃗ on b⃗ is -13/√35 and the direction of the projection vector is same as of b⃗.
Answer and solution
Answer (a) Projection of a⃗ on b⃗ is -13/√35 and the direction of the projection vector…
Solution
Maths sir · 4 steps
Find a⃗·b⃗ first. Its sign gives both the sign of the projection and its direction.
- Dot product. a⃗·b⃗ = (5)(1) + (-1)(3) + (-3)(5) = 5 - 3 - 15 = -13.
- Projection. |b⃗| = √(1 + 9 + 25) = √35. The scalar projection of a⃗ on b⃗ is a⃗·b⃗ divided by |b⃗|.
(a⃗·b⃗)/|b⃗| = -13/√35, projection vector = (a⃗·b⃗)/|b⃗|² b⃗ = -13/35 b⃗
- Direction. The projection vector is a negative multiple of b⃗, so it points opposite to b⃗. That happens because the angle between a⃗ and b⃗ is obtuse.
- So. The projection is -13/√35 and its direction is opposite to b⃗. That is option (a).
The idea to keep. The sign of a⃗·b⃗ decides everything. Positive means the projection lies along b⃗. Negative means it points against b⃗, and the scalar projection carries that minus sign.
Answer from the NTA final answer key. Solution by PrepWiser.