Let the plane $a x+b y+c z+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at the right angles. If $a, b, c, d$...

A vector $\vec{a}$ has components $3 p$ and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about...

Let $P$ be a plane $l x+m y+n z=0$ containing the line, $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$. IF plane $P$ divides the line segment $A B$ joining points $A(-3,-6,1)$...

Let the position vectors of two points $P$ and $Q$ be $3 \hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}+2 \hat{j}-4 \hat{k}$, respectively. Let $R$ and $S$ be two...

Let $(\lambda, 2,1)$ be a point on the plane which passes through the point $(4,-2,2)$. If the plane is perpendicular to the line joining the...

If for $\mathrm{a}>0$, the feet of perpendiculars from the points $\mathrm{A}(\mathrm{a},-2 \mathrm{a}, 3)$ and $\mathrm{B}(0,4,5)$ on the plane $l \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$ are points $\mathrm{C}(0,-\mathrm{a},-1)$ and D...

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