Question #144751
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
1
Expert's answer
2020-11-17T16:41:00-0500

By definition, the projection of the line segment joining (1,2,1),(4,2,1)(1,2,-1), (4,2,1) on the line x2=y1=z1\frac{x}{2}=\frac{y}{-1}=\frac{z}{1} is a absolute value of the projection of the vector a=(4,2,1)(1,2,1)=(3,0,2)\overline{a}=(4,2,1)-(1,2,-1)=(3,0,2) on the vector b=(2,1,1)\overline{b}=(2,-1,1) which is parallel to the above line. Therefore, prba=abb=32+0(1)+214+1+1=86.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)(1,2,-1), (4,2,1) on the line x2=y1=z1\frac{x}{2}=\frac{y}{-1}=\frac{z}{1} is 72\frac{7}{2}.


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