Vector Algebra

Working with vectors will be important when navigating crystal lattices. It is important that you recall the form and construction of these vectors as well as 1.) how to calculate the length of a vector, 2.) how to test for orthogonality between two vectors and 3.) how to calculate the angle between two vectors.

We'll be working in Cartesian coordinate system using an orthonormal basis set. The basis vectors are (where the " $\hat{}$ " symbol denotes a unit vector):

\begin{equation} \boldsymbol{\hat{\textbf{x}}}= (1,0,0) \end{equation}

\begin{equation} \boldsymbol{\hat{\textbf{y}}}= (0,1,0) \end{equation}

\begin{equation} \boldsymbol{\hat{\textbf{z}}}= (0,0,1) \end{equation}

Any three-dimensional Euclidean vector $\mathbf{a}$ can then be expressed as:

\begin{equation} \mathbf{a} = a_1\boldsymbol{\hat{\textbf{x}}} + a_2\boldsymbol{\hat{\textbf{y}}}+ a_3\boldsymbol{\hat{\textbf{z}}} \end{equation}

Or, in column notation:

\begin{equation} \mathbf{a} = \begin{bmatrix} a_1\ a_2\ a_3\ \end{bmatrix} \end{equation}

In this class, we will often be navigating around crystal structures, so we'll use crystallographic convention, in which notation for a lattice vector is condensed to $[uvw]$. More on that in Figure 2.2.2.

You should know how to add and subtract vectors. For example, the addition of the vectors $\mathbf{a}$ and $\mathbf{b}$:

\begin{equation} \mathbf{a}+\mathbf{b} = (a_1+b_1)\boldsymbol{\hat{\textbf{x}}} + (a_2+b_2)\boldsymbol{\hat{\textbf{y}}} + (a_3+b_3)\boldsymbol{\hat{\textbf{z}}} \end{equation}

Subtraction is similar, of course.

You should also know how to calculate the length of a vector. This is:

\begin{equation} |\mathbf{a}| = \sqrt{a_1^2+a_2^2+a_3^2} \end{equation}

Or, if you are more comfortable putting this in terms of the dot-product:

\begin{equation} |\mathbf{a}| = \sqrt{\mathbf{a} \cdot \mathbf{a}} \end{equation}

Finally, it's important to calculate the angle (or at least the cosine of an angle) between two vectors, $\mathbf{a}$ and $\mathbf{b}$, which can be done using the definition of the scalar product:

\begin{align} \mathbf{a} \cdot \mathbf{b} &= |\mathbf{a}||\mathbf{b}| \cos{\theta}\ \cos{\theta} &= \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}| }\ \cos{\theta} &= \frac{a_1 b_1 + a_2 b_2 + a_3 b_3}{\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}} \end{align}

When $\mathbf{a} \cdot \mathbf{b} = 0$, $\cos \theta = 1$ and $\theta = \pi/2$ or $90 ^{\circ}$. The vectors are orthogonal.