`veca` and `vecb` are two given vectors. On these vectors as adjacent sides a parallelogram is constructed. The vector which is the altitude of the parallelogam and which is perpendicular to `veca` is not equal to

Options

A. `{((veca.vecb))/(|veca|^(2))}veca-vecb`
B. `1/(|veca|^(2)){(veca.vecb)veca-(veca.veca)vecb}`
C. `(vecaxx(vecaxxvecb))/(|veca|^(2))`
D. `(vecaxx(vecbxxveca))/(|vecb|^(2))`

Explanation

To solve the problem, we need to find the vector that represents the altitude of a parallelogram constructed on vectors \(\vec{a}\) and \(\vec{b}\) such that this altitude is perpendicular to \(\vec{a}\).

### Step-by-Step Solution:

1. **Understanding the Vectors**:
Let \(\vec{a}\) and \(\vec{b}\) be two vectors. The parallelogram formed by these vectors has one vertex at the origin and the other vertices at points defined by \(\vec{a}\) and \(\vec{b}\).

2. **Finding the Projection of \(\vec{b}\) onto \(\vec{a}\)**:
The projection of vector \(\vec{b}\) onto vector \(\vec{a}\) is given by the formula:
\[ \text{proj}_{\vec{a}} \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \vec{a} \]
This vector represents the component of \(\vec{b}\) that is in the direction of \(\vec{a}\).

3. **Finding the Altitude Vector**:
The altitude of the parallelogram from the tip of vector \(\vec{b}\) to the line defined by vector \(\vec{a}\) is the vector that is perpendicular to \(\vec{a}\). This altitude vector can be found by subtracting the projection from \(\vec{b}\):
\[ \vec{h} = \vec{b} - \text{proj}_{\vec{a}} \vec{b} \]
Substituting the projection we found in the previous step:
\[ \vec{h} = \vec{b} - \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \vec{a}\right) \]

4. **Condition for Perpendicularity**:
For \(\vec{h}\) to be perpendicular to \(\vec{a}\), we need:
\[ \vec{h} \cdot \vec{a} = 0 \]
This condition ensures that the altitude vector is indeed perpendicular to \(\vec{a}\).

5. **Conclusion**:
The vector that is the altitude of the parallelogram and is perpendicular to \(\vec{a}\) is given by:
\[ \vec{h} = \vec{b} - \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \vec{a}\right) \]
This vector \(\vec{h}\) is not equal to \(\vec{a}\) because it represents the component of \(\vec{b}\) that is orthogonal to \(\vec{a}\).