If a vector `veca` is expressed as the sum of two vectors `vec(alpha)` and `vec(beta)` along and perpendicular to a given vector `vecb` then `vec(beta)` is equal to

Options

A. `((vecaxxvecb)xxvecb)/(|vecb|^(2))`
B. `(vecbxx(vecaxxvecb))/(|vecb|^(2))`
C. `(vecbxx(vecaxxvecb))/(|vecb|)`
D. `{(veca.vecb)/((|vecb|)^(2))}vecb`

Explanation

To solve the problem, we need to express the vector \(\vec{a}\) as the sum of two components: \(\vec{\alpha}\) (which is along the vector \(\vec{b}\)) and \(\vec{\beta}\) (which is perpendicular to \(\vec{b}\)). We are tasked with finding the expression for \(\vec{\beta}\).

### Step-by-Step Solution:

1. **Understand the Components**:
The vector \(\vec{a}\) can be expressed as:
\[ \vec{a} = \vec{\alpha} + \vec{\beta} \]
where \(\vec{\alpha}\) is the component of \(\vec{a}\) along \(\vec{b}\) and \(\vec{\beta}\) is the component of \(\vec{a}\) perpendicular to \(\vec{b}\).

2. **Find the Projection**:
The projection of vector \(\vec{a}\) onto vector \(\vec{b}\) is given by:
\[ \vec{\alpha} = \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]

3. **Substitute \(\vec{\alpha}\) into the Equation**:
Substitute the expression for \(\vec{\alpha}\) back into the equation for \(\vec{a}\):
\[ \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} + \vec{\beta} \]

4. **Rearranging to Find \(\vec{\beta}\)**:
Rearranging the equation to isolate \(\vec{\beta}\):
\[ \vec{\beta} = \vec{a} - \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]

5. **Final Expression for \(\vec{\beta}\)**:
Thus, the expression for \(\vec{\beta}\) is:
\[ \vec{\beta} = \vec{a} - \text{proj}_{\vec{b}} \vec{a} \]
This can also be expressed as:
\[ \vec{\beta} = \vec{a} - \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]