If the projection of the vector î + 2ĵ + k̂ on the sum of the two vectors 2î + 4ĵ - 5k̂ and - λ î + 2ĵ + 3k̂ is 1, then λ is equal to __________.

Solution

Maths sir · 4 steps

Add the two vectors first, then project onto that sum: dot product over its length, set equal to 1.

1. Form the sum. Add the two vectors component by component: b⃗ = (2-λ)î + 6ĵ - 2k̂. This is the vector we project onto.

   proj = (a⃗ · b⃗)/|b⃗| = ((2-λ) + 12 - 2)/√((2-λ)² + 36 + 4) = (12-λ)/√((2-λ)² + 40)

2. Set it to 1. Put 12-λ = √((2-λ)² + 40) and square both sides: 144 - 24λ + λ² = λ² - 4λ + 44. The λ² terms cancel, leaving 100 = 20λ.
3. Check. With λ = 5, the numerator is 12 - 5 = 7 and the length is √(9 + 40) = 7. So the projection is 7/7 = 1, as required.
4. So. λ = 5.

The common slip: -5. Writing the î component of the sum as 2+λ instead of 2-λ gives -5, not 5.

The idea to keep. Projection of a⃗ on b⃗ divides by the length of b⃗, the vector you project onto. After squaring, check that the dot product is positive, because a projection of +1 needs it.

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