Question

Write the vector equation of the straight line passing through the point $(a,b,c)$ and parallel to y-axis.

Answer 1

Since, the given line is parallel to y-axis.
$\therefore$ D.C.'s of the line is $(0,1,0)$
Direction vector of the line is $0\hat{i}+\hat{j}+0\hat{k}$
$\Rightarrow \overrightarrow{b}=0\hat{i}+\hat{j}+0\hat{k}$
Now, the line passes through the point $(a,b,c)$
Position vector of the point is $\overrightarrow{a}=a\hat{i}+b\hat{j}+c\hat{k}$

Therefore, vector equation of the line is given by,
$\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$
$=a\hat{i}+b\hat{j}+c\hat{k}+\lambda (0\hat{i}+\hat{j}+0\hat{k})$
$=a\hat{i}+(b+\lambda )\hat{j}+c\hat{k}$