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 ( $\hat{\imath}+$ $2 \hat{\jmath}+3 \hat{k}$ ) and perpendicular to the vectors $\hat{\imath}+\hat{\jmath}+\hat{k}$ and $2 \hat{\imath}-\hat{\jmath}+\hat{k}$.

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Answer

Let $\bar{a}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}, \bar{b}=\hat{\imath}+\hat{\jmath}+\hat{k}$ and $\bar{c}=2 \hat{\imath}-\hat{\jmath}+\hat{k}$
We know that, $\bar{b} \times \bar{c}$ is perpendicular to both $\bar{b}$ and $\bar{c}$.
$\therefore \quad \bar{b} \times \bar{c}$ is parallel to the required line.
$\quad$$\bar{b} \times \bar{c}=\left|\begin{array}{ccc}\hat{\imath} & \hat{\jmath} & \hat{k} \\ 1 & 1 & 1 \\ 2 & -1 & 1\end{array}\right|=2 \hat{\imath}+\hat{\jmath}-3 \hat{k}$
Thus required line passes through $\bar{a}=2 \hat{\imath}+\hat{\jmath}-3 \hat{k}$ and parallel to $2 \hat{\imath}+\hat{\jmath}-3 \hat{k}$.
$\therefore$ Its equation is $\bar{r}=(2 \hat{\imath}+\hat{\jmath}-3 \hat{k})+\lambda(2 \hat{\imath}+\hat{\jmath}-3 \hat{k})$

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