Question

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

Short answer

The cross product of \(\mathbf{k}\) and \(\mathbf{j}\) is -\mathbf{i}.

Step-by-step solution

Recall the rules for cross product

The right-hand rule is commonly used to remember the results of cross products. It states that if fingers of the right hand are curled from the first vector (\(\mathbf{j}\)) to the second vector (\(\mathbf{k}\)), the thumb points in the direction of the cross product. For unit vectors, \(\mathbf{i} \times \mathbf{j} = \mathbf{k}, \mathbf{j} \times \mathbf{k} = \mathbf{i},\) and \(\mathbf{k} \times \mathbf{i} = \mathbf{j}\).

Find the cross product

Using the rules established earlier, we have \(\mathbf{k} \times \mathbf{j} = -\mathbf{i}\). The negative sign arises because we are taking the cross product in the opposite order. The right-hand-rule shows that \(\mathbf{k} \times \mathbf{j}\) points in the opposite direction to \(\mathbf{i}\).

Sketch the result

Start by drawing a 3D coordinate system. Mark the axes with \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\). Now, \(\mathbf{i}\) lies along the x-axis, and \(\mathbf{k} \times \mathbf{j}\) equals -\mathbf{i}, meaning it points in the opposite direction to \(\mathbf{i}\) along the x-axis.