Question

Find the vector equation of the line joining (1,2,3) and (-3,4,3) and show that it is perpendicular to the z-axis.

Answer 1

$\because$ Vector eq. of the line joining two points with position vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\overrightarrow{r}=\overrightarrow{a}+\lambda (\overrightarrow{b}-\overrightarrow{a})$
So, $\overrightarrow{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda \left\{\right(-3\hat{i}+4\hat{j}+3\hat{k})-(\hat{i}+2\hat{j}+3\hat{k}\left)\right\}i.e.,\overrightarrow{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (-4\hat{i}+2\hat{j})....\left(i\right)$
Now the d.r.'s of line (i) are -4,2,0. Also the d.r.'s of z-axis are 0,0,1.
Since (-4)(0)+(2)(0)+(0)(1)=0 so, clearly line (i) is perpnedicular to the z-axis.
(By using $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$, condition for perpendicular lines).