If vectors `veca and vecb` are two adjacent sides of parallelograsm then the vector representing the altitude of the parallelogram which is perpendicular to `veca` is (A) `vecb+(vecbxxveca)/(|veca|^2)` (B) `(veca.vecb)/(vecb|^2)` (C) `vecb-(vecb.veca)/(|veca|)^2)` (D) `(vecaxx(vecbxxveca))/(vecb|^20`

Options

A. `vecb+(vecbxxveca)/(|veca|^(2))`
B. `(veca.vecb)/(|vecb|^(2))`
C. `vecb-(vecb.veca)/(|veca|^(2))veca`
D. `(vecaxx(vecbxxveca))/(|vecb|^(2))`

Explanation

To solve the problem, we need to find the vector representing the altitude of a parallelogram that is perpendicular to vector \(\vec{a}\), given that \(\vec{a}\) and \(\vec{b}\) are two adjacent sides of the parallelogram.

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a parallelogram with adjacent sides represented by vectors \(\vec{a}\) and \(\vec{b}\).
- We need to find the altitude from point \(B\) to line \(OA\) (which is along vector \(\vec{a}\)).

2. **Define the Altitude**:
- Let \(D\) be the foot of the altitude from point \(B\) to line \(OA\).
- The position vector of point \(D\) can be expressed as \(D = T \vec{a}\), where \(T\) is some scalar multiple.

3. **Express the Vector \(DB\)**:
- The vector \(DB\) can be expressed as:
\[ \vec{DB} = \vec{B} - \vec{D} = \vec{b} - T \vec{a} \]

4. **Condition for Perpendicularity**:
- For the altitude \(DB\) to be perpendicular to \(OA\), the dot product must be zero:
\[ \vec{DB} \cdot \vec{a} = 0 \]
- Substituting for \(\vec{DB}\):
\[ (\vec{b} - T \vec{a}) \cdot \vec{a} = 0 \]

5. **Expand the Dot Product**:
- Expanding the dot product gives:
\[ \vec{b} \cdot \vec{a} - T (\vec{a} \cdot \vec{a}) = 0 \]
- Rearranging this equation, we find:
\[ T = \frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2} \]

6. **Substituting Back to Find \(DB\)**:
- Now substitute \(T\) back into the expression for \(\vec{DB}\):
\[ \vec{DB} = \vec{b} - T \vec{a} = \vec{b} - \frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2} \vec{a} \]

7. **Final Expression for the Altitude**:
- Thus, the altitude vector \(DB\) is given by:
\[ \vec{DB} = \vec{b} - \frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2} \vec{a} \]
- This matches with option (C):
\[ \vec{b} - \frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2} \vec{a} \]

### Conclusion:
The vector representing the altitude of the parallelogram which is perpendicular to \(\vec{a}\) is:
\[ \vec{b} - \frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2} \vec{a} \]