Question

Projection of line segment joining $\left(\right.2,3,4\left.\right)$ and $\left(\right.5,6,7\left.\right)$ on plane $2x+y+z=1$ is :

• A. $2$
• B. $\sqrt{3}$
• C. $3$
• D. $3\sqrt{3}$

Answer 1

Answer: (B)

Let $A\equiv \left(\right.2,3,4\left.\right)$ and $B=\left(\right.5,6,7\left.\right).$
Then $AB=3\sqrt{3}$

Plane: $2x+y+z=1$
Projection of $\overset{ \rightarrow }{A B}$ on vector $2\hat{i}+\hat{j}+\hat{k}$ which is
normal to plane,
$BC=\frac{\left(\right. 3 \hat{i} + 3 \hat{j} + 3 \hat{k} \left.\right) \cdot \left(\right. 2 \hat{i} + \hat{j} + \hat{k} \left.\right)}{\sqrt{6}}=\frac{12}{\sqrt{6}}=2\sqrt{6}$
$\therefore $ Projection of $\overset{ \rightarrow }{A B}$ on plane
$A C=\sqrt{(3 \sqrt{3}-(2 \sqrt{6}}=\sqrt{3}$