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...

In a triangle ABC , if $|\overrightarrow{\mathrm{BC}}|=3,|\overrightarrow{\mathrm{CA}}|=5$ and $|\overrightarrow{\mathrm{BA}}|=7$, then the projection of the vector $\overrightarrow{\mathrm{BA}}$ on $\overrightarrow{\mathrm{BC}}$ is equal to

If the distance of the point $(1,-2,3)$ from the plane $x+2 y-3 z+10=0$ measured parallel to the line, $\frac{x-1}{3}=\frac{2-y}{m}=\frac{z+3}{1}$ is $\sqrt{\frac{7}{2}}$, then the value of...

Let $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}+5 \hat{k}$ If $\vec{r} \times \vec{a}=\vec{b} \times \vec{r} \cdot(a \hat{i}+2 \hat{j}+\hat{k})=3$ and $...