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.