Question

If $a.i = 4,$ then $(a.j) \times (2j - 3k)$ is equal to?
$
  1)12 \\
  2)2 \\
  3)0 \\
  4) - 12 \\
 $

Answer 1

Hint: Here we will take the cross product of the given expression by using the properties of the dot and the cross product and accordingly simplify the expression and place the given value in it for the resultant required value.

Complete step-by-step answer:
Take the given expression: $(a.j) \times (2j - 3k)$
Take the multiplicative distribution property in the above expression –
$ = (a.(j \times (2j - 3k))$
Again, apply the additive multiplicative property inside the bracket –
$ = (a.(2(j \times j) - 3(j \times k))$
Now, use the identity $j \times j = 0$ and $j \times k = i$ and place in the above equation –
$ = (a.(2(0) - 3(i))$
When zero is multiplied with any number it gives zero as the resultant value.
$ = - 3a(i))$
The above expression can be re-written as –
$ = - 3(a.i)$
Place the given known in the above expression –
$ = - 3(4)$
Simplify the above expression by finding the product of the terms. When you multiply the positive term and the negative term the resultant term will be negative.
$ = - 12$
Therefore, $(a.j) \times (2j - 3k) = ( - 12)$
So, the correct answer is “Option 4”.

Note: Cross-product can be defined as the vector which is the binary operation on two vectors with the three dimensions. Be careful about the sign convention while finding the product of the terms with different and same signs. When you find the product of two negative or two positive terms the resultant value will be in positive while finding the product of the two terms with different signs then the resultant value will be negative.