The angle between `vec(A)+vec(B)` and `vec(A)xxvec(B)` is

Options

A. 0
B. `pi//4`
C. `pi//2`
D. `pi`

Explanation

To find the angle between the vectors \(\vec{A} + \vec{B}\) and \(\vec{A} \times \vec{B}\), we can follow these steps:

### Step 1: Understand the relationship between the vectors
We need to find the angle \(\theta\) between the vector \(\vec{A} + \vec{B}\) and the vector \(\vec{A} \times \vec{B}\). The angle between two vectors can be determined using the dot product formula:
\[ \cos \theta = \frac{\vec{p} \cdot \vec{q}}{|\vec{p}| |\vec{q}|} \]
where \(\vec{p} = \vec{A} + \vec{B}\) and \(\vec{q} = \vec{A} \times \vec{B}\).

### Step 2: Calculate the dot product
We compute the dot product \((\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B})\):
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B}) = \vec{A} \cdot (\vec{A} \times \vec{B}) + \vec{B} \cdot (\vec{A} \times \vec{B}) \]
Using the property of the dot product, we know that the dot product of a vector with a vector that is perpendicular to it is zero. Therefore:
\[ \vec{A} \cdot (\vec{A} \times \vec{B}) = 0 \]
\[ \vec{B} \cdot (\vec{A} \times \vec{B}) = 0 \]
Thus, we have:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B}) = 0 + 0 = 0 \]

### Step 3: Conclude the angle
Since the dot product is zero, it implies that the angle \(\theta\) between the two vectors is:
\[ \cos \theta = 0 \]
This means:
\[ \theta = 90^\circ \]

### Final Answer
The angle between \(\vec{A} + \vec{B}\) and \(\vec{A} \times \vec{B}\) is \(90^\circ\).

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