Resolved part of vector `veca` and along vector `vecb " is " veca1` and that prependicular to `vecb " is " veca2` then `veca1 xx veca2` is equl to

Options

A. `((vecaxxvecb).vecb)/(|vecb|^(2))`
B. `((veca.vecb)veca)/(|veca|^(2))`
C. `((veca.vecb)(vecbxxveca))/(|vecb|^(2))`
D. `((veca.vecb)(vecbxxveca))/(|vecbxxveca|)`

Explanation

To solve the problem, we need to find the cross product of two vectors, \( \vec{a_1} \) and \( \vec{a_2} \), where \( \vec{a_1} \) is the component of vector \( \vec{a} \) along vector \( \vec{b} \), and \( \vec{a_2} \) is the component of vector \( \vec{a} \) that is perpendicular to \( \vec{b} \).

### Step-by-Step Solution:

1. **Find \( \vec{a_1} \)**:
The component of vector \( \vec{a} \) along vector \( \vec{b} \) can be calculated using the formula:
\[ \vec{a_1} = \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right) \vec{b} \]

2. **Find \( \vec{a_2} \)**:
The component of vector \( \vec{a} \) that is perpendicular to vector \( \vec{b} \) can be calculated as:
\[ \vec{a_2} = \vec{a} - \vec{a_1} \]
Substituting \( \vec{a_1} \) from step 1:
\[ \vec{a_2} = \vec{a} - \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right) \vec{b} \]

3. **Compute \( \vec{a_1} \times \vec{a_2} \)**:
Now, we need to find the cross product \( \vec{a_1} \times \vec{a_2} \):
\[ \vec{a_1} \times \vec{a_2} = \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \right) \times \left( \vec{a} - \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right) \vec{b} \right) \]

4. **Simplify the Cross Product**:
Using the distributive property of the cross product:
\[ \vec{a_1} \times \vec{a_2} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \left( \vec{b} \times \vec{a} \right) - \frac{(\vec{a} \cdot \vec{b})^2}{|\vec{b}|^4} \left( \vec{b} \times \vec{b} \right) \]
Since \( \vec{b} \times \vec{b} = 0 \), we have:
\[ \vec{a_1} \times \vec{a_2} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \left( \vec{b} \times \vec{a} \right) \]

5. **Final Result**:
Therefore, the result of \( \vec{a_1} \times \vec{a_2} \) is:
\[ \vec{a_1} \times \vec{a_2} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} (\vec{b} \times \vec{a}) \]