If `hat(a) and hat(b)` are the unit vectors along `vec(a) and vec(b)` respectively, then what is the projection of `vec(b) ` on `vec(a)`?

Options

A. `vec(a).vec(b)`
B. `hat(a).hat(b)`
C. `hat(a).vec(b)`
D. `|vec(a)xxvec(b)|`

Explanation

To find the projection of vector \(\vec{b}\) on vector \(\vec{a}\), we can use the formula for the projection of one vector onto another. The formula for the projection of vector \(\vec{b}\) onto vector \(\vec{a}\) is given by:

\[ \text{proj}_{\vec{a}} \vec{b} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|^2} \vec{a} \]

### Step-by-Step Solution:

1. **Identify the Unit Vectors**:
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors along \(\vec{a}\) and \(\vec{b}\), we have:
\[ \|\vec{a}\| = 1 \quad \text{and} \quad \|\vec{b}\| = 1 \]

2. **Substitute the Magnitudes**:
The magnitude of \(\vec{a}\) is 1, so \(\|\vec{a}\|^2 = 1^2 = 1\).

3. **Apply the Projection Formula**:
Substitute \(\|\vec{a}\|^2\) into the projection formula:
\[ \text{proj}_{\vec{a}} \vec{b} = \frac{\vec{a} \cdot \vec{b}}{1} \vec{a} = (\vec{a} \cdot \vec{b}) \vec{a} \]

4. **Final Result**:
Thus, the projection of \(\vec{b}\) on \(\vec{a}\) is:
\[ \text{proj}_{\vec{a}} \vec{b} = (\vec{a} \cdot \vec{b}) \vec{a} \]