Question

The length of the projection of the line segment joining the points $(5, -1, 4)$ and $(4, -1, 3)$ on the plane, $x + y + z = 7$ is:

• A. $\frac{2 }{\sqrt{3}}$
• B. $\frac{2 }{3}$
• C. $\frac{1}{3}$
• D. $\sqrt{\frac{2 }{3}}$

Answer 1

Answer: (D)

$\sqrt{\frac{2 }{3}}$

Answer 2

Normal to the plane $x+y+z=7$ is $\vec{n}=\hat{i}+\hat{j}+\hat{k}$ $\overrightarrow{A B}=-\hat{i}-\hat{k} \Rightarrow|\overrightarrow{A B}|=A B=\sqrt{2}$ $B C=\text { Length of projection of } \overrightarrow{A B} \text { on } \vec{n}=|\overrightarrow{A B} \cdot \hat{n}|$ $=\left|(-\hat{i}-\hat{k}) \cdot \frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}\right|=\frac{2}{\sqrt{3}}$ Length of projection of the line segment on the plane is $A C$ $A C^{2}=A B^{2}-B C^{2}=2-\frac{4}{3}=\frac{2}{3}$ $A C^{2}=\sqrt{\frac{2}{3}}$