Question

Find the vector equation of the passing through the point $2i+3j+k$ and parallel to the vector $4i-2j+3k$.

Answer 1

The line passes through the point,

Say $\overrightarrow{a}=2\hat{i}+3\hat{j}+\hat{k}$

And parallel to the line,

say $\overrightarrow{b}=4\hat{i}-2\hat{j}+3\hat{k}$

The vector equation of the line passing through apoint and parallel to the given line is,

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

Substituting the values, we get,

$\overrightarrow{r}=\left(2\hat{i}+3\hat{j}+\hat{k}\right)+\lambda \left(4\hat{i}-2\hat{j}+3\hat{k}\right)$

$\overrightarrow{r}=\left(2+4\lambda \right)\hat{i}+\left(3-2\lambda \right)\hat{j}+\left(1+3\lambda \right)\hat{k}$