Let $P$ be $a$ plane containing the line $\frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2}$ and parallel to the line $\frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7}$. If the point $(1,-1, \alpha)$ lies on the plane P...

Let the mirror image of the point $(1,3, a)$ with respect to the plane $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})-\mathrm{b}=0$ be $(-3,5$, 2). Then the value of la...

If the equation of plane passing through the mirror image of a point $(2,3,1)$ with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and containing the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z+1}{1}$ is...

Let O be the origin. Let $\overline{\mathrm{OP}}=x \hat{\mathrm{i}}+y \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overline{\mathrm{PQ}}=\sqrt{20} \overline{\mathrm{OQ}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 x \hat{\mathrm{k}}, x, y \in \mathrm{R}, x>0$, be such that $\overline{\mathrm{PQ}}=\sqrt{20}$ and...

Let $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ be two non-zero vectors perpendicular to each other and $|\overline{\mathrm{a}}|=|\overline{\mathrm{b}}|$. If $|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|=|\overline{\mathrm{a}}|$, then the angle between the vectors $...

If the equation of the plane passing through the line of intersection of the planes $2 x-7 y+4 z-3= 0,3 x-5 y+4 z+11=0$ and the...

If $\bar{a}=\alpha \hat{i}+b \hat{j}+3 \hat{k}, \bar{b}=\beta \hat{i}-\alpha \hat{j}-\hat{k}$ and $\bar{c}=\hat{i}-2 \hat{j}-\hat{k}$ such that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=1$ and $\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=-3$ then $...

In a triangle ABC, if $|\overrightarrow{\mathrm{BC}}|=8,|\overrightarrow{\mathrm{CA}}|=7,|\overrightarrow{\mathrm{AB}}|=10$, then the projection of the vector $\overrightarrow{\mathrm{AB}}$ on $\overrightarrow{\mathrm{AC}}$ is equal to:

For $\mathrm{p}>0$, a vector $\overrightarrow{\mathrm{v}}_2=2 \hat{\mathrm{i}}+(\mathrm{p}+1) \hat{\mathrm{j}}$ is obtained by rotating the vector $\overrightarrow{\mathrm{v}}_1=\sqrt{3} p \hat{\mathrm{i}}+\hat{\mathrm{j}}$ by an angle $\theta$ about origin in counter clockwise...

Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and, $\vec{b}, \vec{a} \cdot \vec{c}=7$ and $\vec{b}$ is perpendicular to...