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Question

(A) The vector equation of the plane passing through three non-collinear points with position vectors $a, b, c$ is given by $r \cdot (a 	imes b + b \

Vedclass · Accepted Answer

$\frac{[a, b, c]}{|a 	imes b + b 	imes c + c 	imes a|}$

Vedclass · Answer

(A) The vector equation of the plane passing through three non-collinear points with position vectors $a, b, c$ is given by $r \cdot (a 	imes b + b 	imes c + c 	imes a) = [a, b, c]$.This equation is in the form $r \cdot n = p$,where $n = a 	imes b + b 	imes c + c 	imes a$ is the normal vector to the plane and $p = [a, b, c]$.The length of the perpendicular from the origin to the plane $r \cdot n = d$ is given by $\frac{|d|}{|n|}$.Substituting the values,the length of the perpendicular is $\frac{|[a, b, c]|}{|a 	imes b + b 	imes c + c 	imes a|}$.Since the scalar triple product $[a, b, c]$ represents the volume of the parallelepiped formed by vectors $a, b, c$,and the denominator represents the magnitude of the normal vector,the correct expression is $\frac{[a, b, c]}{|a 	imes b + b 	imes c + c 	imes a|}$.

The length of the perpendicular from the origin to the plane passing through three non-collinear points $a, b, c$ is

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