Problem on finding a unit vector
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Find a unit vector in the direction of $\langle 1, 1 \rangle$.
Solution
Recall that
Finding the unit vector in the same direction as another vector
The find the unit vector in the same direction as $\bfx$, divide by the length of $\bfx$.
Let $\bfx = \langle 1, 1\rangle$. Recall that
Definition of vector length
The length of $\langle x_1, x_2\rangle$ is $\sqrt{x_1^2 + x_2^2}$.
We thus compute that $$\left | \langle 1, 1 \rangle \right | = \sqrt{1^2 + 1^2} = \sqrt{2}.$$
Component view of vector-scalar multiplication
Hence the unit vector in the direction of $\langle 1, 1 \rangle$ is $$\frac{\langle 1, 1 \rangle}{\sqrt{2}} = \left \langle \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right \rangle.$$
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