Q22DQ - University Physics with Modern Physics

Question

Let A represent any nonzero Which of the following are legitimate mathematical operations: (a)  $\overrightarrow{A}⦁\left(\overrightarrow{B}-\overrightarrow{C}\right)$ 2; (b)   $\left(\overrightarrow{B}-\overrightarrow{C}\right)\times \overrightarrow{A}$; (c)  $\overrightarrow{A}⦁\left(\overrightarrow{B}\times \overrightarrow{C}\right)$ ; (d)  $\overrightarrow{A}\times \left(\overrightarrow{B}\times \overrightarrow{C}\right)$ ; (e)  $\overrightarrow{A}\times \left(\overrightarrow{B}⦁\overrightarrow{C}\right)$ ? In each case, give the reason for your answer.

Step-by-Step Solution

Answer

(e)   $\overrightarrow{A}\times \left(\overrightarrow{B}⦁\overrightarrow{C}\right)$is not a legitimate vector operation.

Step 1: Legitimate operation

The operations that follow the elementary algebra of vectors are legitimate vector operations.

Step 2: Validation of vector.

(a)  $\overrightarrow{A}⦁\left(\overrightarrow{B}-\overrightarrow{C}\right)$

Here $\left(\overrightarrow{B}-\overrightarrow{C}\right)$ is a vector quantity and dot product takes place between two vectors. Thus,  this is a legitimate operation.

 

(b)  $\left(\overrightarrow{B}-\overrightarrow{C}\right)\times \overrightarrow{A}$

Here $\left(\overrightarrow{B}-\overrightarrow{C}\right)$  is a vector quantity and cross product takes place between two vectors. Thus,  this is a legitimate operation.

 

(c)  $\overrightarrow{A}⦁\left(\overrightarrow{B}\times \overrightarrow{C}\right)$

Here $\left(\overrightarrow{B}\times \overrightarrow{C}\right)$ is a vector quantity and a dot product takes place between two vectors. Thus,  this is a legitimate operation 

 

(d) $\overrightarrow{A}\times \left(\overrightarrow{B}\times \overrightarrow{C}\right)$

Here  $\left(\overrightarrow{B}\times \overrightarrow{C}\right)$  is a vector quantity and cross-product take place between two vectors. Thus,  this is a legitimate operation.

 

(e)  $\overrightarrow{A}\times \left(\overrightarrow{B}⦁\overrightarrow{C}\right)$

Here  $\left(\overrightarrow{B}⦁\overrightarrow{C}\right)$ is a scalar quantity and cross-product take place between two vectors. Thus,  this is not a legitimate operation.

 

Therefore,  (e) is not a legitimate vector operation.