If `vec(a)=2hat(i)+hat(j)-2hat(k)`, then `|vec(a)|=` ____________.

Answer

To find the magnitude of the vector \(\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}\), we follow these steps:

1. **Identify the components of the vector**:
The vector \(\vec{a}\) has components:
- \(x = 2\) (coefficient of \(\hat{i}\))
- \(y = 1\) (coefficient of \(\hat{j}\))
- \(z = -2\) (coefficient of \(\hat{k}\))

2. **Use the formula for the magnitude of a vector**:
The magnitude \(|\vec{a}|\) of a vector \(\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}\) is given by:
\[ |\vec{a}| = \sqrt{x^2 + y^2 + z^2} \]

3. **Substitute the components into the formula**:
Substitute \(x = 2\), \(y = 1\), and \(z = -2\) into the formula:
\[ |\vec{a}| = \sqrt{(2)^2 + (1)^2 + (-2)^2} \]

4. **Calculate the squares of the components**:
- \( (2)^2 = 4 \)
- \( (1)^2 = 1 \)
- \( (-2)^2 = 4 \)

5. **Add the squares**:
\[ |\vec{a}| = \sqrt{4 + 1 + 4} = \sqrt{9} \]

6. **Find the square root**:
\[ |\vec{a}| = 3 \]

Thus, the magnitude of the vector \(\vec{a}\) is \(3\).