Question

Prove the property of the cross product. $$ \mathbf{u} \times \mathbf{u}=\mathbf{0} $$

Short answer

The cross product of a vector by itself results in the zero vector. This is due to the fact that every cross multiplication of identical components will result in 0, giving a final cross product of 0.

Step-by-step solution

Understanding Cross Product

Cross product, also known as vector product, gives a vector which is perpendicular to the vectors being multiplied. The result vector has a direction according to the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.

Express the vectors

Since \(\mathbf{u}\) is a single vector, it's feasible to label it's components as \(u_i\), \(u_j\), and \(u_k\) for the x, y, and z dimensions respectively. Therefore, \(\mathbf{u} = u_i\hat{i} + u_j\hat{j} + u_k\hat{k}\). The cross product is then, \(\mathbf{u} \times \(\mathbf{u}\), which is \(u_i\hat{i} + u_j\hat{j} + u_k\hat{k}\) cross \(u_i\hat{i} + u_j\hat{j} + u_k\hat{k}\).

Compute the cross product

Cross product is normally computed using the determinant of a special 3x3 matrix, but it can also be computed component by component. According to the properties of the cross products, \(i \times i = j \times j = k \times k = 0\). Therefore, every cross multiplication of identical components will result in 0, and as a result, the entire cross product will be 0.