Question

What is the result of $(\overrightarrow A + \overrightarrow B ) \times (\overrightarrow A - \overrightarrow B )$?
A) $2(\overrightarrow A \times \overrightarrow B )$
B) $(\overrightarrow A \times \overrightarrow B )$
C) $2(\overrightarrow A .\overrightarrow B )$
D) $2(\overrightarrow B \times \overrightarrow A )$

Answer 1

A vector in physics is defined as a physical quantity that has both magnitude and direction. It is written by an arrow or hat. The length of the vector is equal to the magnitude of the quantity given and the direction is also the same as that of the quantity given. These quantities are important in studying the motion of the objects. Some examples of vector quantities are force, displacement, momentum and acceleration etc.

Complete step by step solution:
The formula for $\overrightarrow A \times \overrightarrow B = \left| {\overrightarrow A } \right|\left| {\overrightarrow B } \right|\sin \theta \widehat n$
Where $\widehat n$is a unit vector that is perpendicular to the direction of both the vectors. But it can be written that
$\overrightarrow A \times \overrightarrow B = - (\overrightarrow B \times \overrightarrow A )$---(i)
Now $(\overrightarrow A + \overrightarrow B ) \times (\overrightarrow A - \overrightarrow B ) = \overrightarrow A \times \overrightarrow A - \overrightarrow A \times \overrightarrow B + \overrightarrow B \times \overrightarrow A - \overrightarrow B \times \overrightarrow B$---(ii)
If two vectors are in the same direction, then the angle between them will be zero. So it can be written that
$\Rightarrow \overrightarrow A \times \overrightarrow A = \left| {\overrightarrow A } \right|\left| {\overrightarrow A } \right|\sin 0° \widehat n$
$\Rightarrow \overrightarrow A \times \overrightarrow A = 0$
Similarly it can be written that,
$\Rightarrow \overrightarrow B \times \overrightarrow B = 0$
Therefore, equation (ii) can be written as
$\Rightarrow (\overrightarrow A + \overrightarrow B ) \times (\overrightarrow A - \overrightarrow B ) = 0 - (\overrightarrow A \times \overrightarrow B ) + (\overrightarrow B \times \overrightarrow A ) - 0$
$\Rightarrow (\overrightarrow A + \overrightarrow B ) \times (\overrightarrow A - \overrightarrow B ) = - (\overrightarrow A \times \overrightarrow B ) - (\overrightarrow A \times \overrightarrow B )$
$\Rightarrow (\overrightarrow A + \overrightarrow B ) \times (\overrightarrow B \times \overrightarrow A ) = - 2(\overrightarrow A \times \overrightarrow B )$
The result of cross product of vectors is
$\Rightarrow (\overrightarrow A \times \overrightarrow B ) \times (\overrightarrow B \times \overrightarrow A ) = 2(\overrightarrow B \times \overrightarrow A )$

Option D is the right answer.

Note: It is important to note that the vectors are multiplied by using two methods- cross product and dot product. In the scalar product of two vector quantities, the resultant is a scalar value whereas in the vector product of two quantities, the resultant obtained is vector in nature. If the angle between the cross product of two vector quantities is zero, then both the quantities are collinear. If one of the vector quantities is zero or if both the vectors are parallel to each other, then the cross product of both the vectors is not defined.