This problem requires finding a unit vector that has the same direction as the given vector $\mathbf{a} = -3\mathbf{i} + 8\mathbf{j}$.

Steps to solve:

1. Find the magnitude (length) of the vector $\mathbf{a}$: The formula for the magnitude of a vector $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}$ is: $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2}$ Here, $a_1 = -3$ and $a_2 = 8$. Substituting these values: $|\mathbf{a}| = \sqrt{(-3)^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73}.$

2. Divide the vector by its magnitude to get the unit vector: The formula for a unit vector $\mathbf{u}$ in the same direction as $\mathbf{a}$ is: $\mathbf{u} = \frac{\mathbf{a}}{|\mathbf{a}|}.$ Substituting $\mathbf{a} = -3\mathbf{i} + 8\mathbf{j}$ and $|\mathbf{a}| = \sqrt{73}$: $\mathbf{u} = \frac{-3\mathbf{i} + 8\mathbf{j}}{\sqrt{73}} = \left(\frac{-3}{\sqrt{73}}\right)\mathbf{i} + \left(\frac{8}{\sqrt{73}}\right)\mathbf{j}.$

3. Simplify or approximate values (if needed): The components of $\mathbf{u}$ are approximately: $\frac{-3}{\sqrt{73}} \approx -0.351, \quad \frac{8}{\sqrt{73}} \approx 0.937.$ Therefore: $\mathbf{u} \approx -0.351\mathbf{i} + 0.937\mathbf{j}.$

Let me know if you'd like a deeper explanation or further assistance with any part of the solution!

Here are 5 related questions for practice:

1. How do you verify if a given vector is a unit vector?
2. What happens to the direction of a vector when converted to a unit vector?
3. How would you find a unit vector in three-dimensional space?
4. What is the unit vector of $\mathbf{a} = 6\mathbf{i} - 2\mathbf{j}$?
5. How do you interpret unit vectors in physics or engineering applications?

Tip:

Always double-check your magnitude calculations to ensure accuracy when normalizing vectors.