A perpendicular is drawn from a point on the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1}$ to the plane $x+y+z=3$ such that the foot of the perpendicular Q also lies...

If $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}$, then the value of $|\hat{i} \times(\vec{a} \times \hat{i})|^2+|\hat{j} \times(\vec{a} \times \hat{j})|^2+|\hat{k} \times(\vec{a} \times \hat{k})|^2$ is equal to

Let $A$ be a point on the line $\vec{r}=(1-3 \mu) \hat{i}+(\mu-1) \hat{j}+(2+5 \mu) \hat{k}$ and $B(3,2,6)$ be a point in the space. Then the value...

Let $\vec{a}=2 \hat{i}+\lambda_1 \hat{j}+3 \hat{k}, \vec{b}=4 \hat{i}+\left(3-\lambda_2\right) \hat{j}+6 \hat{k}$ and $\vec{c}=3 \hat{i}+6 \hat{j}+\left(\lambda_3-1\right) \hat{k}$ be three vectors such that $\vec{b}=2 \vec{a}$ and $\vec{a}$ is perpendicular...

The plane passing through the point $(4,-1,2)$ and parallel to the lines $\frac{x+2}{3}=\frac{y-2}{-1}=\frac{z+1}{2}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3}$ also passes through the point:

If the distance between the plane, $23 x-10 y-2 z+ 48=0$ and the plane containing the lines $$ \frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3} $$ and $\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda}(\lambda \in R) \quad$...

The equation of the plane passing through the line of intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ and $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})+4=0$ and parallel to the...

If the equation of a plane P , passing through the intersection of the planes $x+4 y-z+7=0$ and $3 x +y+5 z=8$ is $...

Let $x_0$ be the point of local maxima of $f(x)=\vec{a} \cdot(\vec{b} \times \vec{c})$, where $\vec{a}=x \hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=-2 \hat{i}+x \hat{j}+\hat{k}$ and $...