To find a vector $\vec{v}$ that is parallel to $\vec{u} = -\hat{i} + 6\hat{j} - 3\hat{k}$ and has a magnitude of $|\vec{v}| = 4$, we can follow these steps:

1. Calculate the magnitude of $\vec{u}$: $|\vec{u}| = \sqrt{(-1)^2 + 6^2 + (-3)^2} = \sqrt{1 + 36 + 9} = \sqrt{46}.$

2. Find the unit vector in the direction of $\vec{u}$: The unit vector $\hat{u}$ in the direction of $\vec{u}$ is given by: $\hat{u} = \frac{\vec{u}}{|\vec{u}|} = \frac{-\hat{i} + 6\hat{j} - 3\hat{k}}{\sqrt{46}}.$

3. Scale the unit vector to have a magnitude of 4: Since $\vec{v}$ is parallel to $\vec{u}$, we can write $\vec{v} = k \hat{u}$ where $k$ is chosen so that $|\vec{v}| = 4$. Thus, $\vec{v} = 4 \cdot \frac{-\hat{i} + 6\hat{j} - 3\hat{k}}{\sqrt{46}}.$

4. Simplify the expression: Distribute the $4$ in the numerator: $\vec{v} = \frac{-4\hat{i} + 24\hat{j} - 12\hat{k}}{\sqrt{46}}.$

So, the answer is: $\vec{v} = \frac{-4\hat{i} + 24\hat{j} - 12\hat{k}}{\sqrt{46}}.$

Would you like more details on any of these steps, or do you have further questions?

1. How do we calculate the magnitude of a vector?
2. What is a unit vector and why is it useful?
3. Why is $\vec{v}$ parallel to $\vec{u}$ if it is a scalar multiple of $\vec{u}$?
4. How do we determine the scalar factor to adjust vector magnitude?
5. Can this method be applied to any vector with a specified magnitude?

Tip: A vector is parallel to another if it’s a scalar multiple of it, which keeps their directions aligned.