Find the vector equation of the line passing through the point ha… - Vidyadip

Question

Find the vector equation of the line passing through the point having position vector

$\hat{i}+2 \hat{j}+3 \hat{k}$ and perpendicular to vectors $\hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}-\hat{j}+\hat{k}$.

✓

Answer

The vector perpendicular to the vectors $\bar{b}$ and $\bar{c}$ is given

by

$\bar{b} \times \bar{c}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 2 & -1 & 1\end{array}\right|$

$\begin{aligned} & =\hat{i}(1+1)-\hat{j}(1-2)+\hat{k}(-1-2) \\ & =2 \hat{i}+\hat{j}-3 \hat{k}\end{aligned}$

Since the line is perpendicular to the vector $\bar{b}$ and $\bar{c}$, it is parallel to $\bar{b} \times \bar{c}$. The vector

equation of the line passing through $A(\bar{a})$ and parallel to $\bar{b} \times \bar{c}$ is

$\bar{r}=\bar{a}+\lambda(\bar{b} \times \bar{c})$, where $\lambda$ is a scalar.

Here, $\bar{a}=\hat{i}+2 \hat{j}+3 \hat{k}$

Hence, the vector equation of the required line is

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

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