Problem on vector direction and addition
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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:
Building a vector from its length and direction
\begin{align}
\mathbf{x} &= |\mathbf{x} | \text{ dir } \mathbf{x} \\
\mathbf{y} &= | \mathbf{y} | \text{ dir } \mathbf{y}
\end{align}
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
Length of a scalar vector multiplication
For any scalar $c$ and vector $\mathbf{v}$, $|c \mathbf{v} | = |c|\ |\mathbf{v} |$, where $|c|$ is the absolute value of $c$.
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
Direction of a vector
The direction of a vector is itself a unit vector. That is, $|\text{dir } \mathbf{x} | = 1$.
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.
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