What is the length of the perpendicular drawn | vedclass

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(C) The plane passes through point $a$ and contains the line $r = b + \lambda c$. The direction of the line is $c$ and the vector connecting $a$ to $b

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) The plane passes through point $a$ and contains the line $r = b + \lambda c$. The direction of the line is $c$ and the vector connecting $a$ to $b$ is $(b - a)$.The normal vector $n$ to the plane is given by the cross product of the two vectors lying in the plane:$n = (b - a) 	imes c = b 	imes c - a 	imes c = b 	imes c + c 	imes a$.The equation of the plane is $(r - a) \cdot n = 0$,which simplifies to $r \cdot n = a \cdot n$.Substituting $n = b 	imes c + c 	imes a$,we get $r \cdot (b 	imes c + c 	imes a) = a \cdot (b 	imes c + c 	imes a) = [a, b, c]$.The length of the perpendicular from the origin $(0, 0, 0)$ to the plane $r \cdot n = d$ is given by $\frac{|d|}{|n|}$.Here,$d = [a, b, c]$ and $n = b 	imes c + c 	imes a$.Thus,the length is $\frac{[a, b, c]}{|b 	imes c + c 	imes a|}$.

What is the length of the perpendicular drawn from the origin to the plane passing through point $a$ and containing the line $r = b + \lambda c$?

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