The line of intersection of the planes overli | vedclass

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Question

(D) The line of intersection of two planes is perpendicular to the normal vectors of both planes. Let the normal vectors be $\vec{n}_1 = 3 \hat{i}-\ha

Vedclass · Accepted Answer

$-2 \hat{i}+7 \hat{j}+13 \hat{k}$

Vedclass · Answer

(D) The line of intersection of two planes is perpendicular to the normal vectors of both planes. Let the normal vectors be $\vec{n}_1 = 3 \hat{i}-\hat{j}+\hat{k}$ and $\vec{n}_2 = \hat{i}+4 \hat{j}-2 \hat{k}$.The direction vector $\vec{v}$ of the line of intersection is given by the cross product of the normal vectors: $\vec{v} = \vec{n}_1 	imes \vec{n}_2$.$\vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & -1 & 1 \ 1 & 4 & -2 \end{vmatrix}$.$\vec{v} = \hat{i}((-1)(-2) - (1)(4)) - \hat{j}((3)(-2) - (1)(1)) + \hat{k}((3)(4) - (-1)(1))$.$\vec{v} = \hat{i}(2 - 4) - \hat{j}(-6 - 1) + \hat{k}(12 + 1)$.$\vec{v} = -2 \hat{i} + 7 \hat{j} + 13 \hat{k}$.Thus,the line is parallel to the vector $-2 \hat{i} + 7 \hat{j} + 13 \hat{k}$.

The line of intersection of the planes $\overline{r} \cdot(3 \hat{i}-\hat{j}+\hat{k})=1$ and $\overline{r} \cdot(\hat{i}+4 \hat{j}-2 \hat{k})=2$ is parallel to which of the following vectors?

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