Question

A vectar $\vec{n}$ is inclined to x-axis at $45^{\circ}$, toy-axis at $60^{\circ}$ and at an acute angle to z-axis. If $\vec{n}$ is a normal to a plane passing through the point $\left(\sqrt{2}, -1, 1\right)$ then the equation of the plane is :

• A. $4\sqrt{2}x +7y+z-2$
• B. $2x +y+2z=2\sqrt{2}+1$
• C. $3\sqrt{2}x-4y-3z = 7$
• D. $\sqrt{2}x-y-z = 2$

Answer 1

Answer: (B)

Direction cosines of $\overrightarrow{n}$ are $\frac{1}{2}, \frac{1}{4}, \frac{1}{2}$.
Equation of the plane,
$\frac{1}{2}\left(x-\sqrt{2}\right)+\frac{1}{4}\left(y+1\right)+\frac{1}{2}\left(z-1\right) = 0$
$\Rightarrow \quad2\left(x-\sqrt{2}\right) + \left(y + 1\right) + 2\left(z-1\right) = 0$
$\Rightarrow \quad2x+y + 2z= 2\sqrt{2}-1 + 2$
$\Rightarrow \quad2x+y + 2z= 2\sqrt{2}+1$