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

Answer 1

The correct option is C $\sqrt{\frac{2}{3}}$

Here $PQ$ is the projection

Also $PQ=BC$

$AC=\overline{AB}.\overline{BC}=(\hat{i}+\hat{k}).\left(\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}\right)$

$=2/\sqrt{3}$

Also, $AB=\sqrt{2}$

$AB=A{C}^2+B{C}^2=2=4/3+B{C}^2$

$B{C}^2=2/3$

$BC=\sqrt{2/3}$