Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3} \hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the origin in counterclockwise direction in...

Consider the three planes $$ \begin{aligned} & P_1: 3 x+15 y+21 z=9 \\ & P_2: x \cdot 3 y \cdot z=5, \text { and }...

If vectors $\vec{a}_1=x \hat{i}-\hat{j}+\hat{k}$ and $\vec{a}_2=\hat{i}+y \hat{j}+z \hat{k}$ are collinear, then a possible unit vector parallel to the vector $x \hat{i}+y \hat{j}+z \hat{k}$ is

If $(1,5,35),(7,5,5),(1, \lambda, 7)$ and $(2 \lambda, 1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is

The distance of the point $(1,1,9)$ from the point of intersection of the line $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}$ and the plane $x+y+z=17$ is

Let $\vec{a}=\hat{i}+2 \hat{j}-\hat{k}, \vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a}=\vec{c} \times \vec{a}$ an $...

The equation of the plane passing through the point (1, 2, –3) and perpendicular to the planes 3x + y – 2z = 5 and...

If $\vec{a} \quad$ and $\vec{b}$ are perpendicular, then $\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})))$ is equal to.

Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+\mathrm{m}-\mathrm{n}=0$ and $l^2+\mathrm{m}^2-\mathrm{n}^2 =0$. Then then value of $...

The equation of the line through the point $(0,1,2)$ and perpendicular to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is: