JEE Main & Advanced Mathematics Three Dimensional Geometry Foot of Perpendicular from a Point to The Line

Foot of perpendicular from a point \[A(\alpha ,\,\,\beta ,\,\,\gamma )\]to the line  \[\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}\] : If P be the foot of perpendicular, then P is \[(lr+{{x}_{1}},\,mr+{{y}_{1}},\,nr+{{z}_{1}})\]. Find the direction ratios of AP and apply the condition of perpendicularity of AP and the given line. This will give the value of r and hence the point P, which is foot of perpendicular.

Length and equation of perpendicular : The length of the perpendicular is the distance AP and its equation is the line joining two known points A and P.

The length of the perpendicular is the perpendicular distance of given point from that line.

Reflection or image of a point in a straight line : If the perpendicular PL from point P on the given line be produced to Q such that \[PL=QL,\] then Q is known as the image or reflection of P in the given line. Also, L is the foot of the perpendicular or the projection of P on the line.