The length of the perpendicular drawn from the point (2,1,4) to the plane containing the l…

Question

The length of the perpendicular drawn from the point $(2,1,4)$ to the plane containing the lines $\vec r=\hat i+\hat j+\lambda(\hat i+2\hat j-\hat k)$ and $\vec r=\hat i+\hat j+\mu(-\hat i+\hat j-2\hat k)$ is:

Accepted Answer

Vector of the plane is

$$\left| {\begin{pmatrix} {\hat i} & {\hat j} & {\hat k}  \
   1 & 2 & { - 1}  \
   { - 1} & 1 & { - 2}  \

\end{pmatrix} } \right| =  - 3\hat i + 3\hat j + 3\hat k$$

Now equation of plane is

$- 3x + 3y + 3z = c$

(1, 1, 0) will satisfy the plane

$$ \Rightarrow  - 3 + 3 + 0 = c$$

$$ \Rightarrow $$ c = 0

$- 3x + 3y + 3z = 0$

distance  from (2, 1, 4) is

$$ \Rightarrow \left| {{{ - 6 + 3 + 12} \over {\sqrt {27} }}} \right| = \left| {{9 \over {3\sqrt 3 }}} \right| = \sqrt 3 \,\,units$$

Solutions · 1