A Short Treatise on a Vectors Concept

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

\displaystyle \Pi_1: r \cdot n_1 = a \cdot n_1 and \displaystyle \Pi_2: r \cdot n_2 = a \cdot n_2

where n1 and n2 denote the characteristic normals of the two planes \Pi_1 and \Pi_2 respectively, and a common point A with position vector a which lies on both planes.

If the line L with equation  r = a + \lambda(n_1 \times n_2) is a solution to both \Pi_1 and \Pi_2, ie L is the line of intersection of both planes, then we have to show that

(1): \displaystyle \left[ a + \lambda(n_1 \times n_2) \right] \cdot n_1 = a \cdot n_1

and

(2): \displaystyle \left[ a + \lambda(n_1 \times n_2) \right] \cdot n_2 = a \cdot n_2

Before proceeding, recognise that (n_1 \times n_2)\cdot n_1 = (n_1 \times n_2)\cdot n_2 = 0.

For (1), LHS = \left[ a + \lambda(n_1 \times n_2) \right] \cdot n_1 = a\cdot n_1 + \lambda(n_1 \times n_2)\cdot n_1 =a\cdot n_1 = RHS

Similar for (2), LHS = \left[ a + \lambda(n_1 \times n_2) \right] \cdot n_2 = a\cdot n_2 + \lambda(n_1 \times n_2)\cdot n_2 =a\cdot n_2 = 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 n_1 \times n_2 . 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 .

Posted by Kristian

1 comment

Bjarki Ágúst

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 🙂

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