If `vecr.veca=vecr.vecb=vecr.vecc=0 " where "veca,vecb and vecc` are non-coplanar, then

Options

A. `vecrbot(veccxxveca)`
B. `vecrbot(vecaxxvecb)`
C. `vecrbot(vecbxxvecc)`
D. `vecr=vec0`

Explanation

To solve the problem step by step, we start with the given conditions and analyze them logically.

### Step-by-Step Solution:

1. **Given Information**: We have three vectors \( \vec{a}, \vec{b}, \vec{c} \) which are non-coplanar. Additionally, we know:
\[ \vec{r} \cdot \vec{a} = 0, \quad \vec{r} \cdot \vec{b} = 0, \quad \vec{r} \cdot \vec{c} = 0 \]

2. **Understanding the Dot Product**: The dot product \( \vec{r} \cdot \vec{a} = 0 \) implies that vector \( \vec{r} \) is orthogonal (perpendicular) to vector \( \vec{a} \). Similarly, \( \vec{r} \) is also orthogonal to \( \vec{b} \) and \( \vec{c} \).

3. **Implication of Orthogonality**: Since \( \vec{r} \) is orthogonal to three non-coplanar vectors \( \vec{a}, \vec{b}, \vec{c} \), it suggests that \( \vec{r} \) must lie in the plane formed by these vectors. However, since \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar, they cannot all lie in the same plane.

4. **Conclusion from Orthogonality**: If \( \vec{r} \) is orthogonal to three non-coplanar vectors, it implies that \( \vec{r} \) must be the zero vector. This is because the only vector that is orthogonal to all vectors in three-dimensional space is the zero vector.

5. **Final Result**: Therefore, we conclude that:
\[ \vec{r} = \vec{0} \]