Question

Write the equation of the line passing through A$(1,2,3)$ and having the direction in $(1,1,1)$ the vector form and also in the symmetric form.

Answer 1

Consider,

$\begin{array}{c}\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k}\\ \\ \overrightarrow{b}=\hat{i}+\hat{j}+\hat{k}\end{array}$

Since, the given line passes through $(1,2,3)$ and its direction ratios are proportional to $1,1,1$

So, its vector equation is

$\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$

$=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}+\hat{j}+\hat{k})$

And its cartesian equation is

$\frac{x-1}{1}=\frac{y-2}{1}=\frac{z-3}{1}$

$x-1=y-2=z-3$