## Problem on finding a vector with given length and direction

Find the vector of length 2 in the direction of $\langle 1,-1 \rangle$.
• ## Solution

Let $\mathbf{x}$ be the vector we seek. We recall the relationship of a vector to its length and direction:
We are given that $| \mathbf{x} |= 2$. We need to compute $\text{dir } \mathbf{x}$ from the information that $\mathbf{x}$ is in the same direction as $\langle 1,-1 \rangle$:
We need to compute the length $|\langle 1, -1 \rangle |$. To do this, recall that
Hence, $| \langle 1, -1 \rangle | = \sqrt{2}$, and
$$\text{dir } \mathbf{x} = \left \langle \frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}} \right \rangle.$$
Thus
\begin{align}
\mathbf{x} &= 2 \cdot \left \langle \frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}} \right \rangle\\
&= \langle \sqrt{2}, -\sqrt{2} \rangle.
\end{align}