` vecP = (2hati - 2hatj +hatk) ` , then find `|vecP|`

Options

A. a unit vector
B. makes an angle `pi//3` with vector `(2hati- 4hatj + 3hatk) `
C. parallel to vector `( -hati + hatj - 1/2 hatk)`
D. perpendicular to vector `3hati + 2hatj - 2hatk`

Explanation

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

### Step-by-Step Solution:

1. **Identify the components of the vector**:
The vector \( \vec{P} \) can be expressed in terms of its components:
\[ \vec{P} = 2\hat{i} - 2\hat{j} + 1\hat{k} \]
Here, the components are \( a = 2 \), \( b = -2 \), and \( c = 1 \).

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

3. **Substitute the components into the formula**:
Substituting \( a = 2 \), \( b = -2 \), and \( c = 1 \) into the magnitude formula:
\[ |\vec{P}| = \sqrt{(2)^2 + (-2)^2 + (1)^2} \]

4. **Calculate the squares of the components**:
\[ |\vec{P}| = \sqrt{4 + 4 + 1} \]

5. **Add the squared values**:
\[ |\vec{P}| = \sqrt{9} \]

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

### Final Answer:
The magnitude of the vector \( \vec{P} \) is \( 3 \).

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