Given two points P and Q we can convert them to the homogeneous coordinate system, compute their cross product and thus get the equation of line passing through them. I am able to verify it for 2D but not for 3D. An illustrating example is given below and require help in interpreting the results for 3D case.
case: 2D
Let P=[1,1] and Q=[2,2]. Then computing their cross product in the homogenous coordinate system gives [1,1,1] crossProduct [2,2,1] = [-1,1,0] which implies line -x + y + 0z = 0 which is nothing but the line y=x and is the correct answer.
case: 3D Does not verify with correct answer
Let P=[3,-1,2] and Q=[-3,0,1]. Then computing their cross product in the homogenous coordinate system gives [3,-1,2,1] crossProduct [-3,0,1,1] = [-1, 16, 2, 5]. But this does not seem to match with the answer obtained using conventional approach shown below,
