MCQ
The vector equation of the line passing through the point having position vector $2 \hat{i}+\hat{j}-3 \hat{k}$ and perpendicular to vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2 \hat{j}-\hat{k}$ is
- A
- B
- C
- D
(c): Let $\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}+2 \hat{j}-\hat{k}$
The vector perpendicular to $\vec{b}$ and $\vec{c}$ is given by
$\begin{aligned}
& \vec{b} \times \vec{c}=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 1 & 1 \\
1 & 2 & -1
\end{array}\right|=\hat{i}(-1-2)-\hat{j}(-1-1)+\hat{k}(2-1) \\
& =-3 \hat{i}+2 \hat{j}+\hat{k}
\end{aligned}$
Since, the line is perpendicular to $\vec{b}$ and $\vec{c}$ therefore, the line will be parallel to $\vec{b} \times \vec{c}$.
$\therefore \quad$ Vector equation of line passing through $2 \hat{i}+\hat{j}-3 \hat{k}$ and parallel to $\vec{b} \times \vec{c}$ is
$\vec{r}=2 \hat{i}+\hat{j}-3 \hat{k}+\lambda(-3 \hat{i}+2 \hat{j}+\hat{k})$
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