Question

If the cartesian equations of a line are $\frac{3-x}{5}=\frac{y+4}{7}=\frac{2z-6}{4}$, write the vector equation for the line

Answer 1

Cartesian equations of a line: $\frac{3-x}{5}=\frac{y+4}{7}=\frac{2z-6}{4}$

Let, $\frac{3-x}{5}=\frac{y+4}{7}=\frac{2z-6}{4}=t$

$\frac{3-x}{5}=t$

or, $x=3-5t$

Similarly,

$y=7t-4$ and $z=2t+3$

Now, $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$

or, $\overrightarrow{r}=(3-5t)\hat{i}+(7t-4)\hat{j}+(2t+3)\hat{k}$

or, $\overrightarrow{r}=(3\hat{i}-4\hat{j}+3\hat{k})+t(-5\hat{i}+7\hat{j}+2\hat{k})$