Practicing Success
Distance of $P(\vec{p})$ from the plane $\vec{r} . \vec{n}=0$ is: |
$|\vec{p} . \vec{n}|$ $\frac{|\vec{p} \times \vec{n}|}{|\vec{n}|}$ $\frac{|\vec{p} . \vec{n}|}{|\vec{n}|}$ None of these |
$\frac{|\vec{p} . \vec{n}|}{|\vec{n}|}$ |
Let $Q(\vec{q})$ be the foot of altitude drawn from P to the plane $\vec{r} . \vec{n}=0$, $\Rightarrow \vec{q}-\vec{p}=\lambda \vec{n}$ $\Rightarrow \vec{q}=\vec{p}+\lambda \vec{n}$ Also $\vec{q} . \vec{n}=0$ $\Rightarrow(\vec{p}+\lambda \vec{n}) . \vec{n}=0$ $\Rightarrow \lambda=-\frac{(\vec{p} . \vec{n})}{|\vec{n}|^2}$ $\Rightarrow \vec{q}-\vec{p}=-\frac{(\vec{p} . \vec{n})}{|\vec{n}|^2} \vec{n}$ ∴ Required distance $=|\vec{q}-\vec{p}|$ $=\frac{|\vec{p} . \vec{n}|}{|\vec{n}|}=|\vec{p} . \hat{n}|$ Hence (3) is correct answer. |