## Problem on vector direction and addition

If $\text{dir } \mathbf{x} = \text{dir } \mathbf{y}$ show that $|\mathbf{x} + \mathbf{y}| = |\mathbf{x}| + | \mathbf{y} |$.
• ## Solution

We are asked to study the length of the sum of two vectors given information of their directions.
We recall the relationship of vectors to their direction and length:
Because we wish to analyze the length of $\mathbf{x} + \mathbf{y}$, we write out $$\mathbf{x} + \mathbf{y} = |\mathbf{x} | \text{ dir } \mathbf{x} + |\mathbf{y} | \text{ dir } \mathbf{y}.$$
Because $\text{dir } \mathbf{x} = \text{dir } \mathbf{y}$, $$\mathbf{x} + \mathbf{y} = \text{dir } \mathbf{x} \ \left( |\mathbf{x}| + |\mathbf{y}| \right)$$
Taking the length of both sides
\begin{align}
|\mathbf{x} + \mathbf{y}| &= | \text{dir } \mathbf{x} \ \left( |\mathbf{x}| + |\mathbf{y}| \right)|
\end{align}
Recalling that
we let $\mathbf{v} = \text{dir } \mathbf{x}$ and $c = |\mathbf{x} | + | \mathbf{y}|$ to get
$$|\mathbf{x} + \mathbf{y}| = | \text{dir } \mathbf{x}| \cdot \Bigl |\left( |\mathbf{x}| + |\mathbf{y}| \right) \Bigr|.$$
We recall that
Thus,
\begin{align}
|\mathbf{x} + \mathbf{y}| &= \Bigl |\left( |\mathbf{x}| + |\mathbf{y}| \right) \Bigr| \\
&= |\mathbf{x}| + |\mathbf{y}|.
\end{align}
The outer absolute value sign can be removed because $|\mathbf{x}| + |\mathbf{y}|$ is always nonnegative.