The equation of the plane containing the line | vedclass

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Question

(A) The required plane passes through a point with position vector $a_1$ and is parallel to the vectors $a_1$ and $a_2$. If $r$ is the position vector

Vedclass · Accepted Answer

$[r, a_1, a_2] = 0$

Vedclass · Answer

(A) The required plane passes through a point with position vector $a_1$ and is parallel to the vectors $a_1$ and $a_2$. If $r$ is the position vector of any point on the plane,then the vectors $(r - a_1)$,$a_1$,and $a_2$ are coplanar. Therefore,their scalar triple product must be zero: $(r - a_1) \cdot (a_1 	imes a_2) = 0$. Expanding this,we get $r \cdot (a_1 	imes a_2) - a_1 \cdot (a_1 	imes a_2) = 0$. Since $[r, a_1, a_2] = r \cdot (a_1 	imes a_2)$ and $[a_1, a_1, a_2] = 0$ (as two vectors are identical),we have $[r, a_1, a_2] = 0$. Hence,the equation of the plane is $[r, a_1, a_2] = 0$.

The equation of the plane containing the lines $r = a_1 + \lambda a_2$ and $r = a_2 + \lambda a_1$ is

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