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Chapter 12: Problem 2

Explain how to find the direction of the cross product \(\mathbf{u} \times \mathbf{v}\)

Short Answer

Expert verified
Question: Explain the process of finding the direction of the cross product of two vectors. Answer: To find the direction of the cross product between two vectors, first calculate their cross product using the given formula, and then apply the right-hand rule to visualize the direction. The right-hand rule involves pointing your right hand's index finger in the direction of the first vector, bending the middle finger in the direction of the second vector, and the thumb pointing in the direction of the cross product. Make sure to consider the order of the vectors, as switching the order gives the opposite direction. The cross product results in a vector that is orthogonal to both input vectors.

Step by step solution

01

Understand the properties of cross products

The cross product of two vectors results in a third vector, which is orthogonal (perpendicular) to both input vectors. The direction of this third vector is determined by the right-hand rule.
02

Review the right-hand rule

The right-hand rule is a convention used in vector analysis to determine the direction of the cross product. To apply the right-hand rule, we simply follow these steps: 1. Point your right hand's index finger in the direction of the first vector, \(\mathbf{u}\). 2. Bend your right hand's middle finger in the direction of the second vector, \(\mathbf{v}\). 3. Your right hand's thumb points in the direction of the cross product, \(\mathbf{u} \times \mathbf{v}\). Note that the order of the vectors is essential: switching the order gives the opposite direction for the resulting vector, since \(\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})\).
03

Calculate the cross product

In order to calculate the cross product of two given vectors, let's say \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\), we use the following formula: $$ \mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1 \end{pmatrix} $$ Calculating the cross product gives us the vector's direction and magnitude. Once we have the cross product, we can then use the right-hand rule to visualize its direction.
04

Apply the right-hand rule to the calculated cross product

Now that you have the cross product vector, you can apply the right-hand rule to determine its direction. Remember to follow the steps mentioned earlier in Step 2. The direction in which your thumb points when properly aligning your right hand represents the direction of the cross product \(\mathbf{u} \times \mathbf{v}\). In conclusion, finding the direction of the cross product between two vectors involves calculating their cross product and then applying the right-hand rule to visualize the direction. It is essential to remember that the cross product results in a vector that is orthogonal to both input vectors, and its direction depends on the order of the input vectors.

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