Dot Product of Unit Vectors

Theorem

Let $\mathbf a$ and $\mathbf b$ be unit vectors.

Then their dot product $\mathbf a \cdot \mathbf b$ is:

$\mathbf a \cdot \mathbf b = \cos \theta$

where $\theta$ is the angle between $\mathbf a$ and $\mathbf b$.

Proof

We have by definition of dot product :

$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \cos \theta$

where $\norm {\mathbf a}$ denotes the length of $\mathbf a$.

From Length of Unit Vector is 1:

$\norm {\mathbf a} = \norm {\mathbf b} = 1$

from which the result follows.

$\blacksquare$

Sources

• 1927: C.E. Weatherburn: Differential Geometry of Three Dimensions: Volume $\text { I }$ ... (previous) ... (next): Introduction: Vector Notation and Formulae: Products of Vectors