Given `vecC = vecA xx vecB and vecD = vecB xx vecA`. What is the angle between `vecC` and `vecD` ?

Options

A. zero
B. `60^(@)`
C. `90^(@)`
D. `180^(@)`

Answer

To solve the problem, we need to find the angle between the vectors \(\vec{C}\) and \(\vec{D}\), where:

\[ \vec{C} = \vec{A} \times \vec{B} \]
\[ \vec{D} = \vec{B} \times \vec{A} \]

### Step 1: Understand the properties of the cross product
The cross product of two vectors \(\vec{A}\) and \(\vec{B}\) is defined such that:

\[ \vec{A} \times \vec{B} = -(\vec{B} \times \vec{A}) \]

This means that the cross product of \(\vec{B}\) and \(\vec{A}\) is equal to the negative of the cross product of \(\vec{A}\) and \(\vec{B}\).

### Step 2: Relate \(\vec{C}\) and \(\vec{D}\)
From the property mentioned above, we can express \(\vec{D}\) in terms of \(\vec{C}\):

\[ \vec{D} = \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) = -\vec{C} \]

### Step 3: Analyze the relationship between \(\vec{C}\) and \(\vec{D}\)
Since \(\vec{D} = -\vec{C}\), it indicates that \(\vec{C}\) and \(\vec{D}\) are in opposite directions.

### Step 4: Determine the angle between \(\vec{C}\) and \(\vec{D}\)
The angle \(\theta\) between two vectors \(\vec{X}\) and \(\vec{Y}\) can be found using the formula:

\[ \cos(\theta) = \frac{\vec{X} \cdot \vec{Y}}{|\vec{X}| |\vec{Y}|} \]

However, since \(\vec{D} = -\vec{C}\), we can directly conclude that the angle between \(\vec{C}\) and \(\vec{D}\) is:

\[ \theta = 180^\circ \]

### Final Answer
Thus, the angle between \(\vec{C}\) and \(\vec{D}\) is \(180^\circ\).