A vector coplanar with the non-collinear vectors `bara and bar b` is

Options

A. `baraxxbarb`
B. `bara+barb`
C. `bara.barb`
D. `baraxx3barb`

Answer

To solve the question, we need to determine which vector is coplanar with the non-collinear vectors \( \vec{A} \) and \( \vec{B} \).

### Step-by-Step Solution:

1. **Understanding Coplanarity**:
Two vectors are said to be coplanar if they lie in the same plane. For vectors \( \vec{A} \) and \( \vec{B} \), any vector that can be expressed as a linear combination of these two vectors will also lie in the same plane.

2. **Analyzing the Options**:
- **Option A: \( \vec{A} \times \vec{B} \)**:
The cross product of two vectors \( \vec{A} \) and \( \vec{B} \) results in a vector that is perpendicular to the plane formed by \( \vec{A} \) and \( \vec{B} \). Therefore, it cannot be coplanar with \( \vec{A} \) and \( \vec{B} \).

- **Option B: \( \vec{A} + \vec{B} \)**:
The sum of the vectors \( \vec{A} \) and \( \vec{B} \) is a vector that lies in the same plane as \( \vec{A} \) and \( \vec{B} \). Thus, \( \vec{A} + \vec{B} \) is coplanar with \( \vec{A} \) and \( \vec{B} \).

- **Option C: \( \vec{A} \cdot \vec{B} \)**:
The dot product of two vectors results in a scalar, not a vector. Since we are looking for a vector, this option cannot be considered.

- **Option D: \( \vec{A} \times \vec{B} \)**:
Similar to option A, this is again the cross product, which is perpendicular to the plane formed by \( \vec{A} \) and \( \vec{B} \). Therefore, it is not coplanar.

3. **Conclusion**:
The only vector that is coplanar with the non-collinear vectors \( \vec{A} \) and \( \vec{B} \) is \( \vec{A} + \vec{B} \).

Thus, the answer is **Option B: \( \vec{A} + \vec{B} \)**.