If the coordinate of P, Q, R and S be (1,2,2), (2,4,0), ( -3, 0, 1) and ( -1, -2 ,2) respectively, Find the projection of RS on PQ.

Solution:

$\begin{array}{l}P{\rm{(1,2,2) = i + 2j + 2k}}\\{\rm{Q(2,4,0)}}\,{\rm{ =  2i + 4j + 0k}}\\{\rm{R( - 3,0,1) =  - 3i + 0j + k}}\\{\rm{S( - 1, - 2,2) =  - i - 2j + 2k}}\\{\rm{Now,}}\\\overrightarrow {{\rm{PQ}}} {\rm{ = }}\overrightarrow {{\rm{OQ}}} {\rm{ - }}\overrightarrow {{\rm{OP}}}  = i + 2j - 2k\\\overrightarrow {{\rm{RS}}} {\rm{ = }}\overrightarrow {{\rm{OS}}} {\rm{ - }}\overrightarrow {OR} {\rm{ = 2i - 2j + k}}\\And,\;\\{\rm{Projection\;of \;}}\overrightarrow {{\rm{RS}}}\; {\rm{ on\; }}\overrightarrow {{\rm{PQ}}} {\rm{  = }}\frac{{\overrightarrow {{\rm{PQ}}} .\overrightarrow {{\rm{RS}}} }}{{|\overrightarrow {{\rm{RS}}} |}}\\ = \frac{{({\rm{2i - 2j + k}}).(i + 2j - 2k)}}{{\sqrt {{1^2} + {2^2} + {2^2}} }}\\ = \frac{{2 - 4 - 2}}{3}\\ =  - \frac{4}{3}\end{array}$