Question

Find the cross product of the unit vectors and sketch your result. $$ \mathbf{j} \times \mathbf{k} $$

Short answer

The cross product of the unit vectors \( \mathbf{j} \) and \( \mathbf{k} \) is the unit vector \( \mathbf{i} \).

Step-by-step solution

Recognize the unit vectors

The given vectors are \( \mathbf{j} \) and \( \mathbf{k} \). In a standard coordinate system, \( \mathbf{j} \) points in the direction of the Y-axis, and \( \mathbf{k} \), along the Z-axis.

Cross product of the unit vectors

The cross product of any two different standard unit vectors \( \mathbf{i}, \mathbf{j} \), and \( \mathbf{k} \) is defined as: \( \mathbf{i} \times \mathbf{j} = \mathbf{k} \), \( \mathbf{j} \times \mathbf{k} = \mathbf{i} \), \( \mathbf{k} \times \mathbf{i} = \mathbf{j} \) Using this, we get the cross product of \( \mathbf{j} \times \mathbf{k} \) is equal to \( \mathbf{i} \).

Graphical representation

In a 3D coordinate system, represent the vectors \( \mathbf{j} \), \( \mathbf{k} \) and the result of the cross product \( \mathbf{i} \). Starting at the origin, \( \mathbf{j} \) can be drawn along the Y-axis and \( \mathbf{k} \) along the Z-axis. The result of their cross product, \( \mathbf{i} \), should be represented as a vector pointing along the X-axis, demonstrating that it is orthogonal to both \( \mathbf{j} \) and \( \mathbf{k} \).