Search any question & find its solution
Question:
Answered & Verified by Expert
A vector $\vec{n}$ is inclined to $X$-axis at $45^{\circ}$, $\mathrm{Y}$-axis at $60^{\circ}$ and at an acute angle to $\mathrm{Z}$-axis. If $\overrightarrow{\mathrm{n}}$ is normal to a plane passing through the point $(-\sqrt{2}, 1,1)$, then equation of the plane is
Options:
Solution:
2020 Upvotes
Verified Answer
The correct answer is:
$\sqrt{2} x+y+z=0$
Let $\vec{n}$ be inclined at angles $\alpha, \beta, \gamma$ to $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ axes respectively.
$\begin{array}{ll}
& \alpha=45^{\circ}, \beta=60^{\circ}, \gamma=? \\
& \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\
\therefore \quad & \frac{1}{2}+\frac{1}{4}+\cos ^2 \gamma=1 \\
\therefore \quad & \cos ^2 \gamma=\frac{1}{4} \\
\therefore \quad & \gamma=60^{\circ} \\
& \overrightarrow{\mathrm{n}}=\cos \alpha \hat{\mathrm{i}}+\cos \beta \hat{\mathrm{j}}+\cos \gamma \hat{\mathrm{k}} \\
\therefore \quad & \overrightarrow{\mathrm{n}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{2} \hat{\mathrm{j}}+\frac{1}{2} \hat{\mathrm{k}}
\end{array}$
$\therefore \quad$ Equation of the required plane is
$\frac{1}{\sqrt{2}}(x+\sqrt{2})+\frac{1}{2}(y-1)+\frac{1}{2}(z-1)=0$
i.e., $\sqrt{2} x+y+z=0$
$\begin{array}{ll}
& \alpha=45^{\circ}, \beta=60^{\circ}, \gamma=? \\
& \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\
\therefore \quad & \frac{1}{2}+\frac{1}{4}+\cos ^2 \gamma=1 \\
\therefore \quad & \cos ^2 \gamma=\frac{1}{4} \\
\therefore \quad & \gamma=60^{\circ} \\
& \overrightarrow{\mathrm{n}}=\cos \alpha \hat{\mathrm{i}}+\cos \beta \hat{\mathrm{j}}+\cos \gamma \hat{\mathrm{k}} \\
\therefore \quad & \overrightarrow{\mathrm{n}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{2} \hat{\mathrm{j}}+\frac{1}{2} \hat{\mathrm{k}}
\end{array}$
$\therefore \quad$ Equation of the required plane is
$\frac{1}{\sqrt{2}}(x+\sqrt{2})+\frac{1}{2}(y-1)+\frac{1}{2}(z-1)=0$
i.e., $\sqrt{2} x+y+z=0$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.