 find a vector in the direction of 
5
𝑖
^
−
𝑗
^
+
2
𝑘
^
5 
i
^
 − 
j
^
​
 +2 
k
^
  that has a magnitude of 8 units

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.

Find a vector in the direction of 5i^ - j^ + 2k^ that has a magnitude of 8 units.

Learn how to find a vector in the direction of 5i - j + 2k with a magnitude of 8 units. This problem demonstrates key vector concepts like magnitude, unit vector calculation, and vector scaling. Suitable for students in Grades 10-12.

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