Angle between Vector Quantities in terms of Direction Cosines

Theorem
Let $\mathbf a$ and $\mathbf b$ be vector quantities embedded in Cartesian $3$-space

Let $\theta$ be the angle between $\mathbf a$ and $\mathbf b$.

Then:
 * $\cos \theta = \lambda_a \lambda_b + \mu_a \mu_b + \nu_a \nu_b$

where $\lambda_a$, $\mu_a$ and $\nu_a$ are the direction cosines of $\mathbf a$ with respect to the $x$-axis, $y$-axis and $z$-axis respectively, and similarly for $\mathbf b$.

Proof
Let $\mathbf r$ be an arbitrary vector quantity embedded in a Cartesian $3$-space.

From Components of Vector in terms of Direction Cosines:

where:
 * $x$, $y$ and $z$ denote the components of $\mathbf r$ in the $\mathbf i$, $\mathbf j$ and $\mathbf k$ directions respectively.


 * $r$ denotes the magnitude of $\mathbf r$.

Hence: