Calculating the Magnitudes of Three Position Vectors in a Cubic Structure | 신승민

Cubic unit-cell diagram showing three position vectors, each with a magnitude of √3a/2

According to Wolfram MathWorld, Cartesian coordinates are rectilinear two- or three-dimensional coordinates. They are also called rectangular coordinates and a special case of curvilinear coordinates. The Cartesian unit vectors $\hat{e}_1$ , $\hat{e}_2$ , and $\hat{e}_3$ point along the three mutually perpendicular coordinate axes. As shown in the figure, the position vectors are:

\begin{aligned} \mathbf{a}_1 &= \frac{a}{2}(-\hat{e}_1+\hat{e}_2+\hat{e}_3)\\ \mathbf{a}_2 &= \frac{a}{2}(\hat{e}_1-\hat{e}_2+\hat{e}_3)\\ \mathbf{a}_3 &= \frac{a}{2}(\hat{e}_1+\hat{e}_2-\hat{e}_3) \end{aligned}

Because the Cartesian unit vectors are orthonormal and each position vector has three components with absolute value $a/2$ , all three have the same magnitude:

|\mathbf{a}_1|=|\mathbf{a}_2|=|\mathbf{a}_3| =\frac{a}{2}\sqrt{1^2+1^2+1^2} =\frac{\sqrt{3}}{2}a

Reference Video: https://www.youtube.com/watch?v=4-axgJJyd9g