To find a vector in the direction of $5\hat{i} - \hat{j} + 2\hat{k}$ with a magnitude of 8 units, we can follow these steps:

1. Find the magnitude of the given vector $\mathbf{v} = 5\hat{i} - \hat{j} + 2\hat{k}$: $\lVert \mathbf{v} \rVert = \sqrt{5^2 + (-1)^2 + 2^2} = \sqrt{25 + 1 + 4} = \sqrt{30}$

2. Normalize the vector by dividing each component of the vector by its magnitude to make it a unit vector: $\hat{\mathbf{v}} = \frac{1}{\sqrt{30}}(5\hat{i} - \hat{j} + 2\hat{k})$

3. Multiply the unit vector by the desired magnitude of 8: $\mathbf{v'} = 8 \times \frac{1}{\sqrt{30}}(5\hat{i} - \hat{j} + 2\hat{k}) = \frac{8}{\sqrt{30}}(5\hat{i} - \hat{j} + 2\hat{k})$

4. Simplify the expression: $\mathbf{v'} = \frac{40}{\sqrt{30}}\hat{i} - \frac{8}{\sqrt{30}}\hat{j} + \frac{16}{\sqrt{30}}\hat{k}$

Thus, the vector in the direction of $5\hat{i} - \hat{j} + 2\hat{k}$ with a magnitude of 8 units is:

$\mathbf{v'} = \frac{40}{\sqrt{30}}\hat{i} - \frac{8}{\sqrt{30}}\hat{j} + \frac{16}{\sqrt{30}}\hat{k}$

Would you like me to convert this into a more simplified form or provide further details?

---

Related Questions:

1. What is the unit vector of $5\hat{i} - \hat{j} + 2\hat{k}$?
2. How do you find the direction cosines of a vector?
3. What is the dot product of two vectors?
4. How do you compute the cross product of two 3D vectors?
5. What is the significance of the magnitude of a vector?

Tip:

To scale any vector to a desired magnitude, always normalize it first before multiplying by the target magnitude.