3D Geometry
Foot of Perpendicular — Projection Length
nta_pyq_2026_jan
Grade None
Question:
Let $(\alpha,\beta,\gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5,4,2)$ on the line $\vec{r}=(-\hat{i}+3\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}-\hat{k})$. Then the length of the projection of the vector $\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$ on the vector $6\hat{i}+2\hat{j}+3\hat{k}$ is:
Step-by-Step Solution
Key Concept: Any point on line: $Q=(-1+2\lambda,3+3\lambda,1-\lambda)$. $\overrightarrow{PQ}\cdot(2,3,-1)=0$: $(-6+2\lambda,3\lambda-1,-1-\lambda)\cdot(2,3,-1)=0\Rightarrow14\lambda-14=0\Rightarrow\lambda=1$.
Projection $=\dfrac{18}{7}$.
Correct Answer: 3