To find the unit vector of $\mathbf{v} = 5 \mathbf{i} - 3 \mathbf{j}$, we follow these steps:

Step 1: Find the magnitude of vector $\mathbf{v}$

The magnitude $|\mathbf{v}|$ of the vector $\mathbf{v} = 5 \mathbf{i} - 3 \mathbf{j}$ is given by the formula:

$|\mathbf{v}| = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}$

Step 2: Calculate the unit vector

A unit vector is found by dividing the vector by its magnitude. The unit vector $\mathbf{u}$ in the direction of $\mathbf{v}$ is:

$\mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} = \frac{5 \mathbf{i} - 3 \mathbf{j}}{\sqrt{34}}$

Thus, the unit vector is:

$\mathbf{u} = \frac{5}{\sqrt{34}} \mathbf{i} - \frac{3}{\sqrt{34}} \mathbf{j}$

If you'd like, I can also rationalize the denominators, but this is the simplest form.

Would you like any further details on the steps? Here are some related questions to explore:

1. How is a unit vector used in physics and geometry?
2. What is the significance of a vector's magnitude?
3. Can you find the unit vector of a 3D vector?
4. How does the concept of normalization relate to unit vectors?
5. What happens to the direction of a vector when converted to a unit vector?

Tip: The magnitude of a unit vector is always 1, regardless of its direction.