91Ó°ÊÓ

(II) (a) Show that the cross product of two vectors, $$ \begin{array}{c} \overrightarrow{\mathbf{A}}=A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}+A_{z} \hat{\mathbf{k}}, \text { and } \overrightarrow{\mathbf{B}}=B_{x} \hat{\mathbf{i}}+B_{y} \hat{\mathbf{j}}+B_{z} \hat{\mathbf{k}} \text { is } \\ \overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}=\left(A_{y} B_{z}-A_{z} B_{y}\right) \hat{\mathbf{i}}+\left(A_{z} B_{x}-A_{x} B_{z}\right) \hat{\mathbf{j}} \\ +\left(A_{x} B_{y}-A_{y} B_{x}\right) \hat{\mathbf{k}} \end{array} $$ (b) Then show that the cross product can be written $$ \overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}=\left|\begin{array}{ccc} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ A_{x} & A_{y} & A_{z} \\ B_{x} & B_{y} & B_{z} \end{array}\right| $$ where we use the rules for evaluating a determinant. (Note, however, that this is not really a determinant, but a memory aid.)

Step by step solution

01

Understanding the Cross Product Formula

The cross product of two vectors \( \overrightarrow{\mathbf{A}} = A_x \hat{\mathbf{i}} + A_y \hat{\mathbf{j}} + A_z \hat{\mathbf{k}} \) and \( \overrightarrow{\mathbf{B}} = B_x \hat{\mathbf{i}} + B_y \hat{\mathbf{j}} + B_z \hat{\mathbf{k}} \) results in another vector that is perpendicular to both \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \). The resulting vector is calculated using the determinant method of the matrix formed by the unit vectors \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \) and the corresponding components of vectors \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \).

02

Computing Each Component of the Cross Product

We use the formula for the cross product:\[\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}} = (A_yB_z - A_zB_y)\hat{\mathbf{i}} + (A_zB_x - A_xB_z)\hat{\mathbf{j}} + (A_xB_y - A_yB_x)\hat{\mathbf{k}}.\] Each component is computed by finding the determinant of a 2x2 submatrix excluding the unit vector for that component, as per the right-hand rule.

03

Expressing the Cross Product as a Pseudodeterminant

The cross product \( \overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}} \) can be expressed using a formula that resembles the determinant of a 3x3 matrix:\[\left|\begin{array}{ccc} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ A_{x} & A_{y} & A_{z} \ B_{x} & B_{y} & B_{z} \end{array}\right|,\] where the elements correspond to the unit vectors and components of the two vectors. Solving the determinant by expanding along the first row actually yields the original cross product formula.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vectors

At the core of understanding cross products is the concept of vectors. Vectors are mathematical entities that have both magnitude and direction. In physics and engineering, vectors are used to represent quantities like force, velocity, and displacement. A vector in three-dimensional space is often expressed in terms of the unit vectors \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \), which correspond to the x, y, and z axes, respectively.

When dealing with vectors, you can perform various operations, including addition, subtraction, and the cross product. The cross product itself results in another vector that is perpendicular to the two vectors you started with. This property is especially useful in problems involving rotational forces and torques. The cross product is calculated by evaluating a determinant that incorporates the components of each vector.

Determinant Method

The determinant method is a systematic way to calculate the cross product of two vectors. When you set up the matrix for the cross product, the first row consists of the unit vectors \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \), while the subsequent rows are made up of the components of vectors \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \).

The formula for the cross product, \( \overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}} \), can be expressed using a pseudodeterminant that resembles a typical 3x3 matrix determinant:

• First, label the rows: the top contains the unit vectors, the second contains components of \( \overrightarrow{\mathbf{A}} \), and the third contains components of \( \overrightarrow{\mathbf{B}} \).
• Then, expand the determinant along the first row. For each unit vector, calculate the determinant of the 2x2 submatrix that remains after excluding the row and column of that unit vector.
• The results form the components of the resulting cross product vector.

Understanding the determinant method provides a quick and efficient way to compute the cross product and is an excellent memory aid to ensure accurate calculations.

Right-Hand Rule

An essential aspect of computing cross products is understanding the right-hand rule. The right-hand rule is a simple mnemonic to determine the direction of the resulting vector from a cross product. To apply it:

• Imagine aligning your right hand along the direction of the first vector (\( \overrightarrow{\mathbf{A}} \)).
• Next, curl your fingers towards the direction of the second vector (\( \overrightarrow{\mathbf{B}} \)) through the smallest angle.
• Your thumb will point in the direction of the cross product \( \overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}} \).

This physical method allows you to visualize and verify the orientation of vector operations, especially in three-dimensional space, where visualizing perpendicularity can be challenging. Using the right-hand rule helps avoid mistakes and ensures the computed cross product vector points in the correct direction.