# Dot Product - Direction Cosines

The **direction cosines** are three cosine values of the angles a vector makes with the coordinate axes.

Another way to think of this is to view them as the corresponding components of the unit vector pointing in the same direction.

## Definition

The three direction cosines are called \(l, m,\) and \( n, \) respectively.

For a vector \( \vec{v} = a \,\widehat{i} + b \,\widehat{j} + c \,\widehat{k} \) that subtends angles \(\alpha, \beta, \gamma\) with the \(x\)-, \(y\)-, \(z\)-axes, respectively, we have

\[\begin{align} l &= \cos \alpha = \frac{\vec{v} \cdot \widehat{i}}{\left\| \vec{v} \right\|} = \dfrac{a}{\left\| \vec{v} \right\|}\\\\ m &= \cos \beta = \frac{\vec{v} \cdot \widehat{j}}{\left\| \vec{v} \right\|} = \dfrac{b}{\left\| \vec{v} \right\|}\\\\ n &= \cos \gamma = \frac{\vec{v} \cdot \widehat{k}}{\left\| \vec{v} \right\|}= \dfrac{c}{\left\| \vec{v} \right\|}. \end{align}\]

## Direction Cosines Theorem

\[l^2 + m^2 + n^2 = 1 \]

From the definition, it follows that

\[\begin{align} l &= \frac{a}{\sqrt{a^2 + b^2 + c^2}} \\\\ m &= \frac{b}{\sqrt{a^2 + b^2 + c^2}} \\\\ n &= \frac{c}{\sqrt{a^2 + b^2 + c^2}}. \end{align} \]

This is true because \(\left \| \vec{v} \right \| = \sqrt{a^2 + b^2 + c^2} \) by the Pythagorean theorem. Also, note that the dot product with a unit vector returns the component in the direction of that unit vector.

Squaring up and adding the three equations above, it is not too hard to see that

\[l^2 + m^2 + n^2 = 1. \ _\square\]

## Calculating Angles between Vectors

In a similar way, we could use the dot product to calculate the angle between two vectors.

If \(\overrightarrow{v_1}\) and \(\overrightarrow{v_2}\) are two vectors with an angle of \(\theta\) between them,

\[ \cos \theta = \frac{\vec{v_1} \cdot \vec{v_2}}{\left \| \vec{v_1} \right \| \left \| \vec{v_2} \right \| }.\]

This follows directly from the definition of dot product:

\[\begin{align} \vec{v_1} \cdot \vec{v_2} &= \left \| \vec{v_1} \right \| \left \| \vec{v_2} \right \| \cos \theta \\ \Rightarrow \cos \theta &= \frac{\vec{v_1} \cdot \vec{v_2}}{\left \| \vec{v_1} \right \| \left \| \vec{v_2} \right \| }. \ _\square \end{align}\]

**Cite as:**Dot Product - Direction Cosines.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/dot-product-direction-cosines/