The vector product of two vectors `vec A and vec B` is zero. The scalar product of `vec A and (vec A + vec B)` will be

Options

A. zero
B. `A^(2)`
C. AB
D. `A^(2)+AB`

Explanation

To solve the problem, we need to analyze the given information step by step.

### Step 1: Understand the given information
We are given that the vector product (cross product) of two vectors \( \vec{A} \) and \( \vec{B} \) is zero:
\[ \vec{A} \times \vec{B} = 0 \]

### Step 2: Analyze the implications of the cross product being zero
The cross product of two vectors is given by:
\[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \hat{n} \]
where \( \theta \) is the angle between the two vectors, and \( \hat{n} \) is the unit vector perpendicular to the plane formed by \( \vec{A} \) and \( \vec{B} \).

Since the cross product is zero, we have:
\[ |\vec{A}| |\vec{B}| \sin \theta = 0 \]
Given that \( \vec{A} \) and \( \vec{B} \) are non-zero vectors (as stated in the problem), the only way for this equation to hold true is if:
\[ \sin \theta = 0 \]

### Step 3: Determine the angle \( \theta \)
If \( \sin \theta = 0 \), then \( \theta \) must be either \( 0^\circ \) or \( 180^\circ \). This means that the vectors \( \vec{A} \) and \( \vec{B} \) are parallel to each other.

### Step 4: Compute the scalar product
We need to find the scalar product (dot product) of \( \vec{A} \) and \( (\vec{A} + \vec{B}) \):
\[ \vec{A} \cdot (\vec{A} + \vec{B}) = \vec{A} \cdot \vec{A} + \vec{A} \cdot \vec{B} \]

### Step 5: Calculate each term
1. The dot product \( \vec{A} \cdot \vec{A} \) is:
\[ \vec{A} \cdot \vec{A} = |\vec{A}|^2 \]

2. The dot product \( \vec{A} \cdot \vec{B} \) can be expressed as:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \]
Since \( \theta = 0^\circ \) (or \( 180^\circ \)), we have \( \cos \theta = 1 \) (or \( -1 \)). However, since we are dealing with magnitudes, we can consider:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \]

### Step 6: Combine the results
Putting it all together, we have:
\[ \vec{A} \cdot (\vec{A} + \vec{B}) = |\vec{A}|^2 + |\vec{A}| |\vec{B}| \]

### Final Answer
Thus, the scalar product of \( \vec{A} \) and \( (\vec{A} + \vec{B}) \) is:
\[ |\vec{A}|^2 + |\vec{A}| |\vec{B}| \]