... the magnitude of vector B, and ? is the angle between the vectors A and B. We can also represent the three dimensional components of vectors A and B in the following manner: where x1, y1, z1, x2, y2, and z2 are scalar numbers that can take on any value. The magnitudes of the vectors A and B can be represented in the following manner: By the dot product property of linear algebra, the dot product of the two vectors A and B can also be expressed by the following equation: Equating the two representations of the dot product shown above yields the following relationship: This equation can be rearranged to first solve for cos?, and then to solve for ?: Applying the above expression for the magnitudes of the vectors A and B: We can select the [100] vector to be vector A, and the [111] vector to be vector B. The [100] vector therefore ...

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