Question

If $\overrightarrow{C}=\overrightarrow{A}\times \overrightarrow{B}$ and $\overrightarrow{D}=\overrightarrow{B}\times \overrightarrow{A}$ then what is the angle between $\overrightarrow{C}$ and $\overrightarrow{D}$ is:-
$\begin{align}
  & \left( 1 \right){{30}^{\circ }} \\
 & \left( 2 \right){{60}^{\circ }} \\
 & \left( 3 \right){{90}^{\circ }} \\
 & \left( 4 \right){{180}^{\circ }} \\
\end{align}$

Answer 1

Hint: The cross product of two vectors is defined as the product of magnitude of two vectors and sine of the angle between them and, $\hat{n}$ is a unit vector point perpendicular to the plane of A and B. The cross product of two vectors is distributive. But not commutative. Thus the cross product of two vectors is anti-commutative.

Formula used:
The cross product of two vectors can be described as,
$\overrightarrow{A}\times \overrightarrow{B}=AB\sin \theta \hat{n}$
where, are A and B are the magnitudes of the vectors$\overrightarrow{A}$ and $\overrightarrow{B}$
and $\theta $ is the angle between the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$.

Complete step by step answer:
The cross product of two vectors can be described as,
$\overrightarrow{A}\times \overrightarrow{B}=AB\sin \theta \hat{n}$
where, $\hat{n}$ is a unit vector point perpendicular to the plane of A and B.
The angle between two vectors is in between ${{0}^{\circ }}$ and ${{180}^{\circ }}$.
The cross product of two vectors is distributive,
Hence,
$\overrightarrow{A}\times \left( \overrightarrow{B}+\overrightarrow{C} \right)=\left( \overrightarrow{A}\times \overrightarrow{B} \right)+\left( \overrightarrow{A}\times \overrightarrow{C} \right)$
But not commutative. Thus cross product is anticommutative.
But in fact,
$\left( \overrightarrow{B}\times \overrightarrow{A} \right)=-\left( \overrightarrow{A}\times \overrightarrow{B} \right)$
Here $\overrightarrow{A}\times \overrightarrow{B}$ and $\overrightarrow{B}\times \overrightarrow{A}$ are opposite to each other. Hence, the angle between $\overrightarrow{C}$ and $\overrightarrow{D}$ is 180 degrees.
Therefore option (D) is correct.

Note:
The object that has both magnitude and direction is known as a vector. The pointed arrows are used to indicate a vector. There are different types of vectors such as zero vector, unit vector, position vector, coplanar vector and equal vectors. The direction of these cross products is given by right hand rule. For this the forefinger points in the direction of $\overrightarrow{A}$ and $\overrightarrow{B}$ in the direction of the middle finger. Then the thumb shows the direction of the two vectors.