R D Sharma Solutions for Chapter: Vector or Cross Product, Exercise 1: EXERCISE

if $\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=7$ and $\overrightarrow{a}\times \overrightarrow{b}=3\hat{i}+2\hat{j}+6\hat{k},$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{k}$. Find the value of $k$.

Let $\overrightarrow{a}=4\hat{i}+5\hat{j}-\hat{k}, \overrightarrow{b}=\hat{i}-4\hat{j}+5\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}-\hat{k}$. Find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{c}$ and $\overrightarrow{b}$ and $\overrightarrow{d}.\overrightarrow{a}=21$.

The area of the triangle with vertices, $A(2,3,5), B(3,5,8)$ and $C(2,7,8)$ is $\frac{\sqrt{k}}{2}$ square units.

If $\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k},\overrightarrow{b}=-\hat{i}+\hat{k},\overrightarrow{c}=2\hat{j}-\hat{k}$ are three vectors, if the area of the parallelogram is$\frac{\sqrt{k}}{2} \mathrm{sq}. \mathrm{units}$ having diagonals $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ and $\left(\overrightarrow{b}+\overrightarrow{c}\right)$ then find the value of $k$.

The two adjacent sides of a parallelogram are $2\hat{i}-4\hat{j}+5\hat{k}$ and $\hat{i}-2\hat{j}-3\hat{k}.$ Find the unit vector parallel to one of its diagonals. If its area is $11\sqrt{k}$ square units. Then the value of $k$ is

If the area of the triangle with vertices: $A(1,1,2), B(2,3,5)$ and $C(1,5,5)$ is $\frac{\sqrt{k}}{2}$ square units. Find the value of $k$.

If ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}·\overrightarrow{b}\right|}^2=400$ and $\left|\overrightarrow{a}\right|=5,$ then write the value of $\left|\overrightarrow{b}\right|.$

If $\theta$ is the angle between two vectors $\hat{i}-2\hat{j}+3\hat{k}$ and $3\hat{i}-2\hat{j}+\hat{k}$, then $\sin \theta =\frac{2}{7}\sqrt{k}$. Find the value of $k$.