K. C. Sinha and Abhishek Chandra Solutions for Exercise 2: EXERCISE

The length of the projection of the segment joining $\left(x_1, y_1, z_1\right)$ and $\left(x_2, y_2, z_2\right)$ on the line $\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$ is

The lines $\overrightarrow{r}=\alpha (\hat{i}+\hat{j}+\hat{k})+3\hat{k}$ and $\overrightarrow{r}=\beta (\hat{i}-2\hat{j}+\hat{k})+3\hat{k}$

The shortest distance between the lines $\overrightarrow{r}={\overrightarrow{p}}_1+t{\overrightarrow{q}}_1$ and $\overrightarrow{r}={\overrightarrow{p}}_2+s{\overrightarrow{q}}_2$ is

The distance of the point $A\left(\overrightarrow{a}\right)$ from the line $\overrightarrow{r}=\overrightarrow{b}+t\overrightarrow{c}$ is

The length of perpendicular (in units) from the origin to the line $\overrightarrow{r}=(4\hat{i}+2\hat{j}+4\hat{k})+\lambda (3\hat{i}+4\hat{j}-5\hat{k})$ is

The shortest distance between the lines $\overrightarrow{r}=\overrightarrow{a}+k\overrightarrow{b}$ and $\overrightarrow{r}=\overrightarrow{a}+l\overrightarrow{c}$ is (where $\overrightarrow{b}$ and $\overrightarrow{c}$ are non-collinear)

The vector equation of the line passing through the points $A(3, 4, -7)$ and $B(1, -1, 6)$ is

The acute angle between the lines $\overrightarrow{r}=(4\hat{i}-\hat{j})+5(2\hat{i}+\hat{j}-3\hat{k})$ and $\overrightarrow{r}=(\hat{i}-\hat{j}+2\hat{k})+t(\hat{i}-3\hat{j}+2\hat{k})$ is