Question

if a and b are the two vectors then the value of
\[(\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}-\overrightarrow{b})\]
A- \[2(\overrightarrow{b}\times \overrightarrow{a})\]
B- \[-2(\overrightarrow{b}\times \overrightarrow{a})\]
C- \[(\overrightarrow{b}\times \overrightarrow{a})\]
D- \[(\overrightarrow{a}\times \overrightarrow{b})\]

Answer 1

Hint: Vectors can be multiplied with one other using two product rules- broadly dot product which gives a scalar result and cross-product which gives vector result. Here we are given two vectors and we need to find the cross product. We use the laws of vector algebra.

Step by step answer: The given two vectors are \[(\overrightarrow{a}+\overrightarrow{b})\]and \[(\overrightarrow{a}-\overrightarrow{b})\]
We need to find the cross product, so we use the laws of algebra.
\[
   (\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}-\overrightarrow{b}) \\
  =\overrightarrow{a}\times (\overrightarrow{a}-\overrightarrow{b})+\overrightarrow{b}\times (\overrightarrow{a}-\overrightarrow{b}) \\
  =\overrightarrow{a}\times \overrightarrow{a}-\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{a}-\overrightarrow{b}\times \overrightarrow{b} \\
\]
We know that cross product of a vector with itself gives zero,
\[\Rightarrow 0-\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{a}-0\]
\[=-\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{a}\]
Also, vector product is not commutative, so using law of commutativity, we get,
\[
   (\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}-\overrightarrow{b}) =\overrightarrow{b}\times \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{a} \\
  \Rightarrow 2\overrightarrow{(b}\times \overrightarrow{a}) \\
\]
\[\therefore \]the answer is \[2\overrightarrow{(b}\times \overrightarrow{a})\]

 Hence, the correct option is (B).

Note: While taking either dot product or cross product we have to keep in mind we have to take the angle between the two original vectors.As vector multiplication is not commutative so take care while multiplying the vectors.