Question

How do you do the cross product with \[\widehat{i}\] and \[\widehat{j}\] ?

Answer 1

Hint: We discuss the cross product of two given unit vectors \[\widehat{i}\] and \[\widehat{j}\]. First we cross multiply both the given unit vector and after that we will try to find the required solution.

Complete step by step solution:
A unit vector along any vector is a vector whose magnitude is one and whose direction coincides with the direction of the provided vector.
When we find the cross-product of two-unit vectors, a unit square with a unit area is produced. As a result, the resulting vector must be a unit vector, perpendicular to the two-unit vectors, identical in magnitude but directed in the opposite direction.

Image: 3 dimension of coordinate axes
We need to find the cross product of \[\widehat{i} \times \widehat{j}\] .
\[\widehat{i} \times \widehat{j} = \left|\widehat{ i} \right|\left| \widehat{j }\right|\sin {90^ \circ }\widehat n\]
We know that \[\widehat{i}\], \[\widehat{j}\] are unit vectors. So \[\left|\widehat{ i} \right| = 1\] and \[\left| \widehat {j} \right| = 1\]
\[\widehat {i} \times \widehat{j} = 1\widehat n\]
The perpendicular vector on XY plane is Z. The unit vector along Z axis is k.
\[\widehat{i} \times \widehat{j} = \widehat{k}\]
Note: Students need to take care that the cross product of the same unit vector is zero. i.e., \[\widehat{i} \times\widehat{ i} = \widehat{j} \times \widehat{ j} = 0\] . Also, we need to take care that when we find the cross product of \[\widehat{i},\widehat{j}\] then we write it like \[\widehat{i} \times \widehat{j}\], we unable to write like \[\widehat{j} \times \widehat{i}\] . This \[\widehat{j} \times\widehat{ i}\] gives us different value.