What will be the projection of vector A⃗=î+ĵ+k̂ on vector B⃗=î+ĵ ?

Solution

Physics sir · 3 steps

The projection lies along B, so find its length with a dot product and point it along the unit vector of B.

1. Length along B. The component of A⃗ along B⃗ is (A⃗·B⃗)/|B⃗|. Here A⃗·B⃗ = 1 + 1 + 0 = 2 and |B⃗| = √2, so the length is 2/√2 = √2.
2. Give it direction. Multiply that length by the unit vector B = (î+ĵ)/√2. Together these give the projection vector:

   A⃗_∥ = (A⃗·B⃗)/|B⃗|² B⃗ = 2/2(î+ĵ) = î+ĵ

3. So. The projection of A⃗ on B⃗ is î+ĵ, which is option (b).

The idea to keep. A projection on B⃗ always points along B⃗. Its length is A⃗·B, and you multiply that length by B, the unit vector, never by B⃗ itself.

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