The equation of the line,passing through A(1, | vedclass

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(D) Let the position vector of point $A$ be $\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k}$.Let the two given vectors be $\vec{b} = 2 \hat{i} + \hat{j} -

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

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(D) Let the position vector of point $A$ be $\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k}$.Let the two given vectors be $\vec{b} = 2 \hat{i} + \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} + 3 \hat{j} + 2 \hat{k}$.The direction of the line is perpendicular to both $\vec{b}$ and $\vec{c}$,so it is parallel to $\vec{b} 	imes \vec{c}$.$\vec{b} 	imes \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 1 & -1 \ 1 & 3 & 2 \end{vmatrix} = \hat{i}(2 - (-3)) - \hat{j}(4 - (-1)) + \hat{k}(6 - 1) = 5 \hat{i} - 5 \hat{j} + 5 \hat{k}$.We can take the direction vector as $\vec{v} = \hat{i} - \hat{j} + \hat{k}$ (dividing by $5$).The equation of the line is $\vec{r} = \vec{a} + \lambda \vec{v}$,which is $\vec{r} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$.

The equation of the line,passing through $A(1, 2, 3)$ and perpendicular to the vectors $2 \hat{i} + \hat{j} - \hat{k}$ and $\hat{i} + 3 \hat{j} + 2 \hat{k}$,is

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