Embibe Experts Solutions for Chapter: Three Dimensional Geometry, Exercise 1: Manipur Board-2018

Find the distance of a point $A$ with position vector $\overrightarrow{a}$ from the plane $\overrightarrow{r}·\overrightarrow{n}=q$.

Find the shortest distance between the following pairs of parallel lines $\overrightarrow{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (2\hat{i}+3\hat{j}+4\hat{k})$ and $\overrightarrow{r}=2\hat{i}+4\hat{j}+5\hat{k}+\mu (4\hat{i}+6\hat{j}+8\hat{k})$.

Find the equation of the plane through the point $\left(1,0,-1\right)$ and $\left(3,2,2\right)$ and parallel to the line $x-1=\frac{y-1}{-2}=\frac{z-2}{3}$.

Derive the vector equation of a line passing through a given point and parallel to a given vector and hence obtain the Cartesian equation of the line.

Derive the vector equation of a plane in the normal form and hence obtain the Cartesian equation of the plane.

Find the equation of the plane through the point $\left(1,-1,3\right)$ and parallel to the plane $\overrightarrow{r}·(2\hat{i}+3\hat{j}-4\hat{k})+5=0$.

Find the distance between the two parallel lines $\overrightarrow{r}=(\hat{i}+\hat{j})+\lambda (2\hat{i}+\hat{j}+\hat{k})$ and $\overrightarrow{r}=(2\hat{i}+\hat{j}-\hat{k})+\mu (2\hat{i}+\hat{j}+\hat{k})$.

Obtain the equation of a plane in the intercept form.