Three Dimensional Geometry

The distance of the point $(3,8,2)$ from the line $\frac{x-1}{2}=\frac{y-3}{4}=\frac{z-2}{3}$ measured parallel to the plane $3 x+2 y-2 z+15=0$, is

The image of the point with position vector $\hat{\mathbf{i}}+3 \hat{\mathbf{k}}$ in the plane $\mathbf{r} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=1$ is

If the foot of the perpendicular drawn from the point $(0,2,1)$ on a line passing through $(\alpha, 5,1)$ is $\left(\frac{5}{3}, \frac{7}{3}, \frac{15}{i}\right)$, then $\alpha$ is equal to _____

The image of the point with position vector $i+3 \mathbf{k}$ in the plane $r \cdot(i+j+k)=1$ is

If the foot of the perpendicular drawn from the point $(2,0,1)$ on a line passing through $(\alpha, 5,1)$ is $\left(\frac{7}{3}, \frac{5}{3}, \frac{15}{4}\right)$, then $\alpha$ is equal to_____

If $\left(\frac{1}{2}, \frac{1}{3}, n\right)$ are the direction cosines of a line, then the value of $n$ is

If $\alpha, \beta, \gamma$ are the angles which a half ray makes with the positive direction of the axes, then $\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma$ is equal to

The distance of the point $(1,-5,9)$, from the plane $\mathbf{r} \cdot(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ measured along the line $\mathbf{r}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ is

The direction cosines of any normal to the $x y$-plane are

If a line makes $\alpha, \beta, \gamma$ with the positive direction of $x, y$ and $z$-axes respectively. Then, $\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma$ is equal to

The image of the point $P(1,3,4)$ in the plane $2 x-y+z+3=0$, is

The projection of a line on a co-ordinate axes are $2,3,6$. Then the length of the line is