Question
The length of the projection of the line segment joining the points $$\left( {5,\, - 1,\,4} \right)$$ and $$\left( {4,\, - 1,\,3} \right)$$ 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 :
$$\sqrt {\frac{2}{3}} $$
Solution :
$$\eqalign{ & AC = \overrightarrow {AB} .\widehat {AC} = \left( {\hat i + \hat j} \right).\frac{{\left( {\hat i + \hat j + \hat k} \right)}}{{\sqrt 3 }} = \frac{2}{{\sqrt 3 }} \cr & {\text{Now, }}A'B' = BC = \sqrt {A{B^2} - A{C^2}} \, = \sqrt {2 - \frac{4}{3}} = \sqrt {\frac{2}{3}} \, \cr} $$
$$\therefore $$ Length of projection $$ = \sqrt {\frac{2}{3}} \,$$
$$\eqalign{ & AC = \overrightarrow {AB} .\widehat {AC} = \left( {\hat i + \hat j} \right).\frac{{\left( {\hat i + \hat j + \hat k} \right)}}{{\sqrt 3 }} = \frac{2}{{\sqrt 3 }} \cr & {\text{Now, }}A'B' = BC = \sqrt {A{B^2} - A{C^2}} \, = \sqrt {2 - \frac{4}{3}} = \sqrt {\frac{2}{3}} \, \cr} $$
$$\therefore $$ Length of projection $$ = \sqrt {\frac{2}{3}} \,$$