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Chapter 9: Problem 43

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

Expert verified
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

01

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.
02

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: \((c(u_2v_3 - u_3v_2), c(u_3v_1 - u_1v_3), c(u_1v_2 - u_2v_1))\).
03

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: \((c(u_2v_3 - u_3v_2), c(u_3v_1 - u_1v_3), c(u_1v_2 - u_2v_1))\).
04

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.

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