If the projection of 2î+4ĵ-2k̂ on î+2ĵ+αk̂ is zero. Then, the value of α will be ___________.

Solution

Physics sir · 3 steps

Zero projection means the two vectors are perpendicular, so set their dot product to zero.

1. Projection formula. The projection of A⃗ on B⃗ is (A⃗ · B⃗)/|B⃗|. The magnitude |B⃗| is never zero here, so the projection is zero only when A⃗ · B⃗ = 0.

   (2)(1) + (4)(2) + (-2)(α) = 0

2. Solve for alpha. This gives 2 + 8 - 2α = 0, so 2α = 10 and α = 5.
3. So. The value of α is 5.

The common slip: -5. Dropping the minus sign on the -2k̂ term gives 2 + 8 + 2α = 0, which leads to -5, not 5.

The idea to keep. Whenever a projection or a component of one vector along another is zero, the vectors are perpendicular. Multiply matching components, add them, and set the sum to zero.

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