I have been in contact with Frederick Koh from Whitegroupmaths.com who kindly agreed that he would write a guest post for the blog to promote what he has to offer – Tutoring in A level maths in Singapore. So without further ado let me present you with the real content.
I have been asked on numerous occasions by students to provide a short effective mathematical proof verifying the fact that obtaining the vector product of the normals characterising two separate non parallel planes in 3 dimensional Cartesian space produces the direction vector of the line arising from the intersection of the above mentioned planes. I shall share this here:
(Note that the reader is assumed to possess knowledge of basic scalar and vector product operations)
Editors note: If you are not familiar with this topic, I can recommend some of the vidoes from Khan Academy on linear algebra.
Let the scalar product equations of two non parallel planes be
where n1 and n2 denote the characteristic normals of the two planes and respectively, and a common point A with position vector a which lies on both planes.
If the line L with equation is a solution to both and , ie L is the line of intersection of both planes, then we have to show that
Before proceeding, recognise that .
For (1), LHS = = RHS
Similar for (2), LHS = = RHS
Reconciling the truths of (1) and (2) therefore yields the observation that the direction vector of the line of intersection between two planes is equivalent to . Hope this helps. Peace.
About the Author: Frederick Koh is a teacher residing in Singapore who specialises in teaching the A level maths curriculum. He has accumulated more than a decade of tutoring experience and loves to share his passion for mathematics on his personal site www.whitegroupmaths.com .