Question

Find the vector equation of the line joining $(1,2,3)$ and $(-3,4,3)$ and show pependicular to the z-axis

Answer 1

Vector equation passing through $(1,2,3)(-3,4,3)$ is

$\overline{r}=\overline{a}+\lambda (b-a)$

$a=i+2j+3k,b=-3i+4j+3k$

$\overline{r}=(\overline{i}+2j+3k)+\lambda (-3i+4j+3k-i-2j-3k)$

$\therefore$ Equation of line $=(ii+2j+3k)+\lambda (-4i+2j)$

To prove perpendicular to z-axis.

$a_1{a}_2+b_1{b}_2+c_1{c}_2=0$

$-4i+zj$

$a_1=4,b_1=2,c_1=0$

Direct rations $\Rightarrow 0i+0j+1k$

$a_2=,b_2=0,c_2=1$

$\therefore a_1{a}_2+b_1{b}_2+c_1{c}_2=0+0+0=0$

Hence proved.