Question

The value of \[\overset{\hat{\ }}{\mathop{i}}\,.\left( \overset{\hat{\ }}{\mathop{j}}\,\times \overset{\hat{\ }}{\mathop{k}}\, \right)+\overset{\hat{\ }}{\mathop{j}}\,.\left( \overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{k}}\, \right)+\overset{\hat{\ }}{\mathop{k}}\,.\left( \overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{j}}\, \right)\]
(A) 0
(B) -1
(C) 1
(D) 3

Answer 1

Hint: We are given an expression and we are asked to solve the given expression and find its value. So, we have these unit vectors arranged in an expression with dot product and cross product and we have to find the respective value. We know that, \[\overset{\hat{\ }}{\mathop{i}}\,.\overset{\hat{\ }}{\mathop{i}}\,=\overset{\hat{\ }}{\mathop{j}}\,.\overset{\hat{\ }}{\mathop{j}}\,=\overset{\hat{\ }}{\mathop{k}}\,.\overset{\hat{\ }}{\mathop{k}}\,=1\]. Also, we know that the cross product gives, \[\overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{j}}\,=\overset{\hat{\ }}{\mathop{k}}\,\], \[\overset{\hat{\ }}{\mathop{j}}\,\times \overset{\hat{\ }}{\mathop{k}}\,=\overset{\hat{\ }}{\mathop{i}}\,\] and \[\overset{\hat{\ }}{\mathop{k}}\,\times \overset{\hat{\ }}{\mathop{i}}\,=\overset{\hat{\ }}{\mathop{j}}\,\]. Using these values, we will solve and find the most appropriate value from the options given to us. Hence, we will have the required value.

Complete step by step answer:
According to the given question, we have an expression based on unit vectors which are connected with the dot product and the cross product. We are asked to solve the expression given to us and find the respective value.
The expression we have is,
\[\overset{\hat{\ }}{\mathop{i}}\,.\left( \overset{\hat{\ }}{\mathop{j}}\,\times \overset{\hat{\ }}{\mathop{k}}\, \right)+\overset{\hat{\ }}{\mathop{j}}\,.\left( \overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{k}}\, \right)+\overset{\hat{\ }}{\mathop{k}}\,.\left( \overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{j}}\, \right)\]
We know that, the dot product of similar unit vector gives the value as 1, that is,
\[\overset{\hat{\ }}{\mathop{i}}\,.\overset{\hat{\ }}{\mathop{i}}\,=\overset{\hat{\ }}{\mathop{j}}\,.\overset{\hat{\ }}{\mathop{j}}\,=\overset{\hat{\ }}{\mathop{k}}\,.\overset{\hat{\ }}{\mathop{k}}\,=1\]
Also, the cross product of the unit vectors gives us the value as,
\[\overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{j}}\,=\overset{\hat{\ }}{\mathop{k}}\,\], \[\overset{\hat{\ }}{\mathop{j}}\,\times \overset{\hat{\ }}{\mathop{k}}\,=\overset{\hat{\ }}{\mathop{i}}\,\] and \[\overset{\hat{\ }}{\mathop{k}}\,\times \overset{\hat{\ }}{\mathop{i}}\,=\overset{\hat{\ }}{\mathop{j}}\,\]
We will use these fundamentals to solve the expression, so we have.
Firstly, applying the cross products within the brackets, we get,
\[\Rightarrow \overset{\hat{\ }}{\mathop{i}}\,.\left( \overset{\hat{\ }}{\mathop{i}}\, \right)+\overset{\hat{\ }}{\mathop{j}}\,.\left( \overset{\hat{\ }}{\mathop{-j}}\, \right)+\overset{\hat{\ }}{\mathop{k}}\,.\left( \overset{\hat{\ }}{\mathop{k}}\, \right)\]
Now, applying the dot product values, we get,
\[\Rightarrow \overset{\hat{\ }}{\mathop{i}}\,.\overset{\hat{\ }}{\mathop{i}}\,+\left( -\overset{\hat{\ }}{\mathop{j}}\,.\overset{\hat{\ }}{\mathop{j}}\, \right)+\overset{\hat{\ }}{\mathop{k}}\,.\overset{\hat{\ }}{\mathop{k}}\,\]
Rearranging further, we get,
\[\Rightarrow \overset{\hat{\ }}{\mathop{i}}\,.\overset{\hat{\ }}{\mathop{i}}\,-\overset{\hat{\ }}{\mathop{j}}\,.\overset{\hat{\ }}{\mathop{j}}\,+\overset{\hat{\ }}{\mathop{k}}\,.\overset{\hat{\ }}{\mathop{k}}\,\]
We will now put the respective values and we get the value of the expression as,
\[\Rightarrow 1-1+1=1\]

So, the correct answer is “Option C”.

Note: The unit vectors should be carefully written and taken into computation. Even though order is not important in dot products, but in case of cross product, the order is everything. Change in order means change in signs and if carelessly dealt will lead to wrong answers. If the cross product of \[\overset{\hat{\ }}{\mathop{k}}\,\times \overset{\hat{\ }}{\mathop{i}}\,=\overset{\hat{\ }}{\mathop{j}}\,\], then the cross product of \[\overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{k}}\,=-\overset{\hat{\ }}{\mathop{j}}\,\].