A) \[\left( \frac{5}{3},\frac{7}{3},\frac{17}{3} \right)\]
B) \[(5,7,17)\]
C) \[\left( \frac{5}{7},-\frac{7}{3},\frac{17}{3} \right)\]
D) \[\left( -\frac{5}{3},\frac{7}{3},-\frac{17}{3} \right)\]
E) None of these
Correct Answer: A
Solution :
Let D be the foot of the perpendicular and let it divide BC in the ratio\[\lambda :1\]. Then, the coordinates of Dare \[\left( \frac{3\lambda +4}{\lambda +1},\frac{5\lambda +7}{\lambda +1},\frac{3\lambda +1}{\lambda +1} \right)\]. Now, \[\overrightarrow{AD}\bot \overrightarrow{BC}\Rightarrow \overrightarrow{AD}.\overrightarrow{BC}=0\] \[\Rightarrow \]\[\left[ \frac{(2\lambda +3)}{\lambda +1}\hat{i}+\frac{(5\lambda +7)}{\lambda +1}\hat{j}-\frac{2}{\lambda +1}\hat{k} \right]\] \[(-\hat{i}-2\hat{j}+2\hat{k})\] \[\Rightarrow \] \[-(2\lambda +3)-2(5\lambda +7)-4=0\] \[\Rightarrow \] \[\lambda =-\frac{7}{4}\] Hence, coordinates of D are\[\left( \frac{5}{3},\frac{7}{3},\frac{17}{3} \right)\].
You need to login to perform this action.
You will be redirected in
3 sec
