Linear Algebra and Its Applications, Exercise 3.2.8

Exercise 3.2.8. Consider a tetrahedon representing the methane molecule CH4, with vertices (hydrogen atoms) at \left(0,0,0\right), \left(1,1,0\right), \left(1,0,1\right), and \left(0,1,1\right), and the center (carbon atom) at \left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right). What is the cosine of the angle between the rays going from the center to each of the vertices?

Answer: Let a be the ray from \left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right) to \left(0,0,0\right) and b be the ray from \left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right) to \left(1,1,0\right). We have

a = \left(0,0,0\right) - \left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right) = \left(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right)

b = \left(1,1,0\right) - \left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right) = \left(\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right)

a^Tb = -\frac{1}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot (-\frac{1}{2})

= -\frac{1}{4} - \frac{1}{4} + \frac{1}{4} = -\frac{1}{4}

\|a\| = \sqrt{\left(-\frac{1}{2}\right)^2+\left(-\frac{1}{2}\right)^2+\left(-\frac{1}{2}\right)^2}

= \sqrt{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}

\|b\| = \sqrt{\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^2+\left(-\frac{1}{2}\right)^2}

= \sqrt{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}

We can then compute the cosine of the angle \theta between a and b as

\cos \theta = a^Tb/\|a\|\|b\| = -\frac{1}{4}/\left(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right)

= -\frac{1}{4}/\left(\frac{3}{4}\right) = -\frac{1}{4} \cdot \frac{4}{3} = -\frac{1}{3}

The cosines of the angles between the other rays have the same value; this can be easily demonstrated using similar steps to those shown above.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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