Q2P - Introduction to Electrodynamics

Question

Is the cross product associative? 

       $\mathbf{\text{(A×B)×C=A×(B×C)}}$

If so, prove it; if not, provide a counterexample (the simpler the better).

Step-by-Step Solution

Answer

The values of the two cross product $A\times (B\times C)$  and $(A\times B)\times C$, are not equal. Thus $(A\times B)\times C\ne A\times (B\times C)$.

Step 1: Explain the concept and describe the given information

The cross product is associative if the result of cross products $\mathbf{\text{A×(B×C)}}$ and $\mathbf{\text{(A×B)×C}}$ is same.

Step 2: Assume the vectors

To prove, let us assume three vectors as$A=4i+3j, B=3i+2j$,  and$C=6i+5j$.

Obtain the value of $A\times (B\times C)$ using the vectors assumed.

$$\begin{gathered} A\times (B\times C)=(4i+3j)\times (3i+2j)\times (6i+5j) \\ =(4i+3j)\times (15k-12k) \\ =(4i+3j)\times \left(3k\right) \\ =(12j+9i) ......... \left(1\right) \end{gathered}$$

Obtain the value of $(A\times B)\times C$ using the vectors assumed.

$$\begin{gathered} (A\times B)\times C=\left((4i+3j)\times 3i+2j)\right)\times (6i+5j) \\ =(12k-9k)\times (6i+5j) \\ =\left(3k\right)\times (6i+5j) \\ =18j-15i ......\left(2\right) \end{gathered}$$

Step 3: Compare the values.

It can be concluded from the equation (1) and (2), that the values of the two cross product $A\times (B\times C)$ and $(A\times B)\times C$, are not equal. Thus

$(A\times B)\times C\ne A\times (B\times C)$