If the projection of i+2 j+ k on the sum of 2 i+4 j-5 k and - i+2 j+3 k is 1, then is ____…

Question

If the projection of $\hat i+2\hat j+\hat k$ on the sum of $2\hat i+4\hat j-5\hat k$ and $-\lambda\hat i+2\hat j+3\hat k$ is 1, then $\lambda$ is \_\_\_\_\_.

Answer: 5

Accepted Answer

Sum $=(2-\lambda)\hat i+6\hat j-2\hat k$. Projection condition: $\dfrac{(\hat i+2\hat j+\hat k)\cdot[(2-\lambda)\hat i+6\hat j-2\hat k]}{|(2-\lambda)\hat i+6\hat j-2\hat k|}=1$

$$(2-\lambda)+12-2 = |(2-\lambda,6,-2)| \implies (12-\lambda)^2=(2-\lambda)^2+36+4$$
$$144-24\lambda+\lambda^2 = \lambda^2-4\lambda+4+40 \implies 144-24\lambda=44-4\lambda \implies 100=20\lambda \implies \lambda=5$$

Solutions · 1