Question

Prove the property of the cross product. $$ c(\mathbf{u} \times \mathbf{v})=(c \mathbf{u}) \times \mathbf{v}=\mathbf{u} \times(c \mathbf{v}) $$

Short answer

The property of the cross product, \(c(\mathbf{u} \times \mathbf{v})=(c \mathbf{u}) \times \mathbf{v}=\mathbf{u} \times(c \mathbf{v})\), holds true since the scalar multiplication of the cross product of two vectors is the same as the cross product of the scalar multiplied vectors.

Step-by-step solution

Defining the vectors and scalar

Start by defining two vectors \(\mathbf{u}\) and \(\mathbf{v}\) in three-dimensional space as well as a scalar value \(c\). These vectors can be given, for instance, as \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\). The scalar \(c\) is an arbitrary real number.

Calculate the scalar multiplied cross product

Next, calculate the scalar multiplied cross product \(c(\mathbf{u} \times \mathbf{v})\). Based on the calculation rules of the cross product, the result vector has the components: \((c(u_2v_3 - u_3v_2), c(u_3v_1 - u_1v_3), c(u_1v_2 - u_2v_1))\).

Calculate the cross product of the scalar multiplied vectors

Now, calculate the cross product of the scalar multiplied vectors, \((c \mathbf{u}) \times \mathbf{v}\) and \(\mathbf{u} \times (c \mathbf{v})\). The resulting vectors in these cases would have the components: \((c u_1, c u_2, c u_3) \times (v_1, v_2, v_3)\) and \((u_1, u_2, u_3) \times (c v_1, c v_2, c v_3)\). Using the rules of the cross product, these would yield the vectors: \((c(u_2v_3 - u_3v_2), c(u_3v_1 - u_1v_3), c(u_1v_2 - u_2v_1))\).

Compare results

By comparing the results from Step 2 and Step 3, one can see that the vectors have the same components. Thus, these vectors are equal, which means that the property of the cross product holds.