Answer to Question #144751 in Analytic Geometry for Dhruv Rawat

Question

The projection of the line segment joining (1,2,-1), (4,2,1) on the line x/2=-y=z is 7/2
True or false with correct explanation

Answer 1

By definition, the projection of the line segment joining $(1,2,-1), (4,2,1)$ on the line $\frac{x}{2}=\frac{y}{-1}=\frac{z}{1}$ is a absolute value of the projection of the vector $\overline{a}=(4,2,1)-(1,2,-1)=(3,0,2)$ on the vector $\overline{b}=(2,-1,1)$ which is parallel to the above line. Therefore, $pr_{\overline{b}}\overline a=\frac{|\overline{a}\cdot\overline{b}|}{|\overline{b}|} = \frac{|3\cdot 2+0\cdot(-1)+2\cdot 1|}{\sqrt{4+1+1}}=\frac{8}{\sqrt{6}}.$

We conclude that it is not true that the projection of the line segment joining $(1,2,-1), (4,2,1)$ on the line $\frac{x}{2}=\frac{y}{-1}=\frac{z}{1}$ is $\frac{7}{2}$.