Question

\[[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{a}\times \overrightarrow{b}]\] is equal to
A. \[\left| \overrightarrow{a}\times \overrightarrow{b} \right|\]
B. \[{{\left| \overrightarrow{a}\times \overrightarrow{b} \right|}^{2}}\]
C. $0$
D. None of these

Answer 1

Hint: In the above question, we are asked to calculate the value of the matrix \[[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{a}\times \overrightarrow{b}]\], whose value can be easily calculated using the concept of the magnitude of a vector and the concept of the scalar triple product.

Formula used: The dot product of two vectors is
$\overrightarrow{a}\cdot \overrightarrow{b}=\left| \overrightarrow{a} \right|\left| \overrightarrow{b} \right|\cos (\overrightarrow{a},\overrightarrow{b})$
The cross-product of two vectors is
$\overrightarrow{a}\times \overrightarrow{b}=\left| \overrightarrow{a} \right|\left| \overrightarrow{b} \right|\sin (\overrightarrow{a},\overrightarrow{b})\overrightarrow{n}$

Complete step by step solution: Here, we are asked about the value of the determinant of the given matrix. The value of the determinant can be calculated using the concept of the scalar triple product.
In this concept, there are three vectors given of which any two are cross-multiplied and then the result of that product is multiplied with the other vector through dot multiplication.
Here, in the above question, we can use the same concept. The three given vectors are \[\overrightarrow{a}\], \[\overrightarrow{b}\], and \[\overrightarrow{a}\times \overrightarrow{b}\].
Then, the given vector is
\[\begin{align}
  & [\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{a}\times \overrightarrow{b}]=(\overrightarrow{a}\times \overrightarrow{b})\cdot (\overrightarrow{a}\times \overrightarrow{b}) \\
 & \text{ }={{\left| \overrightarrow{a}\times \overrightarrow{b} \right|}^{2}} \\
\end{align}\]

Thus, Option (B) is correct.

Additional Information: Important vector identities for solving vector equations are:
\[\overrightarrow{a}\times \overrightarrow{a}=0\]
\[[\overrightarrow{a}\text{ }\overrightarrow{a}\text{ }\overrightarrow{b}]=[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{a}]=[\overrightarrow{b}\text{ }\overrightarrow{a}\text{ }\overrightarrow{a}]=0\]
\[\begin{align}
  & \overrightarrow{i}\cdot \overrightarrow{i}=\overrightarrow{j}\cdot \overrightarrow{j}=\overrightarrow{k}\cdot \overrightarrow{k}=1 \\
 & \overrightarrow{i}\times \overrightarrow{j}=\overrightarrow{k} \\
 & \overrightarrow{j}\times \overrightarrow{k}=\overrightarrow{i} \\
 & \overrightarrow{k}\times \overrightarrow{i}=\overrightarrow{j} \\
\end{align}\]

Note: We can see that using the concept of the scalar triple product we can easily calculate the determinant of any matrix in which the vectors which are given are coplanar. And take care of the vector identities too.