let 

a⃗ =2i^−j^+2k^a→=2i^−j^+2k^

 , then find 

a⃗ |a⃗ |

Given the vector \( \vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \), we are tasked with finding the normalized vector, which is denoted as \( \frac{\vec{a}}{|\vec{a}|} \).

### Steps:

1. **Find the magnitude of vector \( \vec{a} \):**

The magnitude of \( \vec{a} \) is given by:
\[
|\vec{a}| = \sqrt{(2)^2 + (-1)^2 + (2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3
\]

2. **Normalize the vector \( \vec{a} \):**

To find the normalized vector \( \frac{\vec{a}}{|\vec{a}|} \), divide each component of \( \vec{a} \) by its magnitude:
\[
\frac{\vec{a}}{|\vec{a}|} = \frac{1}{3}(2\hat{i} - \hat{j} + 2\hat{k})
\]

Thus, the normalized vector is:
\[
\frac{\vec{a}}{|\vec{a}|} = \frac{2}{3}\hat{i} - \frac{1}{3}\hat{j} + \frac{2}{3}\hat{k}
\]

### Final Answer:
\[
\frac{\vec{a}}{|\vec{a}|} = \frac{2}{3}\hat{i} - \frac{1}{3}\hat{j} + \frac{2}{3}\hat{k}
\]

Would you like further details or have any questions?

### Related Questions:
1. What is the significance of normalizing a vector?
2. How do you find the direction cosines of a vector?
3. What are unit vectors and how are they used in physics?
4. How do vector operations like dot product and cross product differ?
5. How does the magnitude of a vector affect its physical meaning in applications?

**Tip:** Always check your vector's magnitude before normalizing to ensure the resulting unit vector is correct.

Learn how to find the normalized vector of a given vector \( \vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \). This problem involves calculating the magnitude of the vector and then normalizing it, covering key concepts in vectors and vector operations. Suitable for students in Grades 10-12.

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