Question

A line passes through the point with position vector $2\hat{i}-3\hat{j}+4\hat{k}$ and is in the direction of $3\hat{i}+4\hat{j}-5\hat{k}.$ Find equations of the line in vector and cartesian form.

Answer 1

We know that the vector equation of a line passing through a point with position vector $\overrightarrow{a}$ and parallel to the vector $\overrightarrow{b}$ is $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$.

Here,
$$\begin{gathered} \overrightarrow{a}=2\hat{i}-3\hat{j}+4\hat{k} \\ \overrightarrow{b}=3\hat{i}+4\hat{j}-5\hat{k} \end{gathered}$$.

So, the vector equation of the required line is
$$\begin{gathered} \overrightarrow{r}=\left(2\hat{i}-3\hat{j}+4\hat{k}\right)+\lambda \left(3\hat{i}+4\hat{j}-5\hat{k}\right)...\left(1\right) \\ \mathrm{Here},\lambda \mathrm{is}\mathrm{a}\mathrm{parameter}. \end{gathered}$$

Reducing (1) to cartesian form, we get

$$\begin{gathered} x\hat{i}+y\hat{j}+z\hat{k}=\left(2\hat{i}-3\hat{j}+4\hat{k}\right)+\lambda \left(3\hat{i}+4\hat{j}-5\hat{k}\right)[\mathrm{Putting}\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}in(1\left)\right] \\ \Rightarrow x\hat{i}+y\hat{j}+z\hat{k}=\left(2+3\lambda \right)\hat{i}+\left(-3+4\lambda \right)\hat{j}+\left(4-5\lambda \right)\hat{k} \\ \mathrm{Comparing}\mathrm{the}\mathrm{coefficients}\mathrm{of}\hat{i},\hat{j}\mathrm{and}\hat{k},\mathrm{we}\mathrm{get} \\ x=2+3\lambda ,y=-3+4\lambda ,z=4-5\lambda \\ \Rightarrow \frac{x-2}{3}=\lambda ,\frac{y+3}{4}=\lambda ,\frac{z-4}{-5}=\lambda \\ \Rightarrow \frac{x-2}{3}=\frac{y+3}{4}=\frac{z-4}{-5}=\lambda \\ \mathrm{Hence},\mathrm{the}\mathrm{cartesian}\mathrm{form}\mathrm{of}\left(1\right)\mathrm{is} \\ \frac{x-2}{3}=\frac{y+3}{4}=\frac{z-4}{-5} \end{gathered}$$