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

and

where n_{1} and n_{2} 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

(1):

and

(2):

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 .

When this was first posted, I didn’t understand the content at all. I didn’t even know what a plane was. Now, after taking a Linear Algebra course, I can even proof this myself. It’s nice to see that my mathematics skills are improving 🙂