[FIXED] Find perpendicular distance (vector) from a point to a straight line in 3D space

Issue

Inspired by the answer here, I would like to calculate the perpendicular distance (in vector format instead of just magnitude) from a point to a straight line in 3D space.

The above mentioned equation does give the magnitude.

import numpy as np
norm = np.linalg.norm

p1 = np.array([0,0,0])
p2 = np.array([10,0,3])

p3 = np.array([6, -3, 5])

ppDist = np.abs(norm(np.cross(p2-p1, p1-p3)))/norm(p2-p1)
print(ppDist) # 4.28888043816146

ppDist vector in reality would be (0.8807339448,3,-2.935779817) such that its norm() is 4.288880438

Here's a quick visualization -

Solution

Find projection of P13 vector on P12 and the vector difference of P13 from the projection would be the perpendicular vector.

P12 = p2-p1
P13 = p3-p1

proj_13over12 = np.dot(P13, P12)*P12/norm(P12)**2 # array([6.88073394, 0.        , 2.06422018])
perpendicular = proj_13over12 - P13

print(perpendicular) # [ 0.88073394  3.         -2.93577982]

Answered By - beta green