Equation of line passing through $\hat{i} + 3 \hat{j} + 2 \hat{k}$ and $⊥$ to the line $\vec{r} = (\hat{i} + 2 \hat{j} - \hat{k}) + λ (2 \hat{i} + \hat{j} + \hat{k})$ and $\vec{r} = (2 \hat{i} + 6 \hat{j} + \hat{k})$ $+ μ (\hat{i} + 2 \hat{j} + 3 \hat{k})$ is

Moderate

A

$\vec{r} = (\hat{i} + 2 \hat{j} - \hat{k}) + λ (- \hat{i} + 5 \hat{j} - 3 \hat{k})$
B

$\vec{r} = \hat{i} + 3 \hat{j} + 2 \hat{k} + λ (\hat{i} - 5 \hat{j} + 3 \hat{k})$
C

$\vec{r} = \hat{i} + 3 \hat{j} + 2 \hat{k} + λ (\hat{i} + 5 \hat{j} + 3 \hat{k})$
D

$\vec{r} = \hat{i} + 3 \hat{j} + 2 \hat{k} + λ (- \hat{i} - 5 \hat{j} - 3 \hat{k})$

Solution

The required line passes through the point $\hat{i} + 3 \hat{j} + 2 \hat{k}$ and is perpendicular to the lines $\vec{r} = (\hat{i} + 2 \hat{j} - \hat{k}) + λ (2 \hat{i} + \hat{j} + \hat{k})$ and
$\vec{r} = (2 \hat{i} + 6 \hat{j} + \hat{k})$ $+ μ (\hat{i} + 2 \hat{j} + 3 \hat{k})$
therefore, it is parallel to the vector $\vec{b} = (2 \hat{i} + \hat{j} + \hat{k}) \times (\hat{i} + 2 \hat{j} + 3 \hat{k}) = (\hat{i} - 5 \hat{j} + 3 \hat{k})$
Hence, the equation of the required line is $\vec{r} = (\hat{i} + 3 \hat{j} + 2 \hat{k}) + λ (\hat{i} - 5 \hat{j} + 3 \hat{k})$

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