To find a unit vector in the same direction as vector $\mathbf{a} = (6, 9, 4)$, we follow these steps:

1. Calculate the Magnitude of $\mathbf{a}$: The magnitude $|\mathbf{a}|$ of vector $\mathbf{a}$ is given by: $|\mathbf{a}| = \sqrt{6^2 + 9^2 + 4^2} = \sqrt{36 + 81 + 16} = \sqrt{133}$ So, $|\mathbf{a}| = \sqrt{133}$.

2. Divide Each Component by the Magnitude: A unit vector in the same direction as $\mathbf{a}$ is: $\mathbf{u} = \frac{\mathbf{a}}{|\mathbf{a}|} = \left( \frac{6}{\sqrt{133}}, \frac{9}{\sqrt{133}}, \frac{4}{\sqrt{133}} \right)$

Thus, the unit vector in the same direction as $\mathbf{a}$ is approximately: $\left( \frac{6}{\sqrt{133}}, \frac{9}{\sqrt{133}}, \frac{4}{\sqrt{133}} \right)$

Let me know if you would like a more detailed explanation, have any questions, or need further clarification!

Related Questions

1. What is the significance of a unit vector?
2. How do we calculate the magnitude of a 3-dimensional vector?
3. Can you find a unit vector in the opposite direction of $\mathbf{a}$?
4. What does it mean geometrically for two vectors to be in the same direction?
5. How can we verify that the vector we found is indeed a unit vector?

Tip: To simplify computations, you can leave square roots in the denominator for exact answers or rationalize if required for specific applications.