Embibe Experts Solutions for Chapter: Applications of Vector Algebra, Exercise 1: Exercise

If $\overrightarrow{a} \text{and} \overrightarrow{b}$ are two vectors such that $\left|\overrightarrow{a}\right|=5, \left|\overrightarrow{b}\right|=4$ and $\left(\overrightarrow{a}·\overrightarrow{b}\right)=10$,  find $\left|\overrightarrow{a}\times \overrightarrow{b}\right|$ and also find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

Test whether the following pair of lines intersect or not.

$\overrightarrow{r}=-\hat{i}-3\hat{j}-5\hat{k}+t(3\hat{i}+5\hat{j}+7\hat{k}), \overrightarrow{r}=2\hat{i}+4\hat{j}+6\hat{k}+t(\hat{i}+3\hat{j}+5\hat{k})$

The point of intersection of the lines $\overrightarrow{r}=(-\hat{i}+2\hat{j}+3\hat{k})+t(-2\hat{i}+\hat{j}+\hat{k})$ and $\overrightarrow{r}=(2\hat{i}+3\hat{j}+5\hat{k})+s(\hat{i}+2\hat{j}+3\hat{k})$ is 

The vector equation of a plane passing through the line of intersection of the planes $\overrightarrow{r}·\overrightarrow{n_1}=q_1\text{ and }\overrightarrow{r}·{\overrightarrow{n}}_2=q_2\text{ is }$

Find the distance between the parallel planes.
$\overrightarrow{r}·(2\hat{i}-\hat{j}+3\hat{k})=4$ and $\overrightarrow{r}·(6\hat{i}-3\hat{j}+9\hat{k})=-13$

Find the Cartesian equation of the plane containing the line $\frac{x-2}{2}=\frac{y-2}{3}=\frac{z-1}{-2}$ and passing through the point $(-1, 1, -1)$.

Find the vector and Cartesian equations of the plane through the points $(1,2,3)$ and $(2,3,1)$ and perpendicular to the plane $3x-2y+4z-5=0$

If the foot of the perpendicular drawn from the origin to the plane is $(4,-2,-5)$, then find the equation of the plane in vector and cartesian form.