## 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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