Question

If $A=2\widehat{i}-3\widehat{j}+7\widehat{k}$, $B=\widehat{i}+2\widehat{k}$ and $C=\widehat{j}-\widehat{k}$. Find $A\cdot \left( B\times C \right)$.

Answer 1

Hint: To solve the given question, we will proceed in parts. First, we will find the cross product of $B$ and $C$, and as it is given in the question that we need to then find the dot product of $A$ and the cross product of $B$ and $C$. After doing these two operations we will reach the final result.

Complete step-by-step answer:
Let us first find the cross product of $B$ and $C$,
$\begin{align}
  & B\times C=\left( \begin{matrix}
   \widehat{i} & \widehat{j} & \widehat{k} \\
   1 & 0 & 2 \\
   0 & 1 & -1 \\
\end{matrix} \right) \\
 & \Rightarrow B\times C=\widehat{i}\left[ 0\times \left( -1 \right)-2\times 1 \right]-\widehat{j}\left[ 1\times \left( -1 \right)-2\times 0 \right]+\widehat{k}\left[ 1\times 1-0\times 0 \right] \\
 & \Rightarrow B\times C=-2\widehat{i}+\widehat{j}+\widehat{k} \\
\end{align}$
Now, let us move to the other step where we have to find the dot product of $A$ with the answer that we got in the previous step:
$\begin{align}
  & A\cdot \left( B\times C \right)=\left( 2\widehat{i}-3\widehat{j}+7\widehat{k} \right)\cdot \left( -2\widehat{i}+\widehat{j}+\widehat{k} \right) \\
 & \Rightarrow A\cdot \left( B\times C \right)=2\times \left( -2 \right)+\left( -3 \right)\times \left( -1 \right)+7\times 1 \\
 & \Rightarrow A\cdot \left( B\times C \right)=-4-3+7 \\
 & \therefore A\cdot \left( B\times C \right)=0 \\
\end{align}$

Additional Information: The quantities that have both, magnitude and direction are known as vector quantities, while the quantities that only contain magnitude are known as scalar quantities. All the mathematical operations can be performed on scalar quantities while there are certain rules for operation for vector quantities. Two of the operations that are performed on vector quantities are dot product and cross product. The dot product is the product of the magnitude of the vectors and their cosine angles; therefore, it results in a scalar quantity. While cross product is a product of the magnitude of the vectors and their sine angles, and thus it results in a vector quantity.

Note: There are two types of products of two vectors, i.e., dot product and cross product. When a dot product of two vectors is calculated it gives a scalar quantity because of which dot product is also known as scalar product, while when cross product of two vectors is calculated it gives a vector quantity.