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Chapter 1: Problem 52

(a) Use vector components to prove that two vectors commute for both addition and the scalar product. (b) Prove that two vectors anticommute for the vector product; that is, prove that \(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}=-\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{A}}\).

Short Answer

Expert verified
Vector addition and scalar product are commutative; vector cross product is anticommutative.

Step by step solution

01

Vector Addition Commutativity

To show that vector addition is commutative, consider vectors \( \boldsymbol{A} = (A_x, A_y, A_z) \) and \( \boldsymbol{B} = (B_x, B_y, B_z) \). The sum of \( \boldsymbol{A} \) and \( \boldsymbol{B} \) is given by \( \boldsymbol{A} + \boldsymbol{B} = (A_x + B_x, A_y + B_y, A_z + B_z) \). Now, if we reverse the order: \( \boldsymbol{B} + \boldsymbol{A} = (B_x + A_x, B_y + A_y, B_z + A_z) \). Since addition of real numbers is commutative, \( A_x + B_x = B_x + A_x \), \( A_y + B_y = B_y + A_y \), and \( A_z + B_z = B_z + A_z \). Thus, \( \boldsymbol{A} + \boldsymbol{B} = \boldsymbol{B} + \boldsymbol{A} \), proving that vector addition is commutative.
02

Scalar Product Commutativity

The scalar product (dot product) of vectors \( \boldsymbol{A} = (A_x, A_y, A_z) \) and \( \boldsymbol{B} = (B_x, B_y, B_z) \) is given by \( \boldsymbol{A} \cdot \boldsymbol{B} = A_x B_x + A_y B_y + A_z B_z \). Similarly, \( \boldsymbol{B} \cdot \boldsymbol{A} = B_x A_x + B_y A_y + B_z A_z \). Since multiplication of real numbers is commutative, \( A_x B_x = B_x A_x \), \( A_y B_y = B_y A_y \), and \( A_z B_z = B_z A_z \). Therefore, \( \boldsymbol{A} \cdot \boldsymbol{B} = \boldsymbol{B} \cdot \boldsymbol{A} \), confirming that the scalar product is commutative.
03

Vector Product Anticommutativity

The vector product (cross product) of vectors \( \boldsymbol{A} = (A_x, A_y, A_z) \) and \( \boldsymbol{B} = (B_x, B_y, B_z) \) is given by \( \boldsymbol{A} \times \boldsymbol{B} = (A_yB_z - A_zB_y, A_zB_x - A_xB_z, A_xB_y - A_yB_x) \). Reversing the order, \( \boldsymbol{B} \times \boldsymbol{A} = (B_yA_z - B_zA_y, B_zA_x - B_xA_z, B_xA_y - B_yA_x) \). Notice each component changes sign: \( -(A_yB_z - A_zB_y) = B_yA_z - B_zA_y, -(A_zB_x - A_xB_z) = B_zA_x - B_xA_z, -(A_xB_y - A_yB_x) = B_xA_y - B_yA_x \). Hence, \( \boldsymbol{A} \times \boldsymbol{B} = - (\boldsymbol{B} \times \boldsymbol{A}) \), proving the vector product is anticommutative.

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Key Concepts

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

Vector Addition
Vector addition is a fundamental operation involving vectors, essential for physics and engineering applications. When you add vectors, you are essentially combining their respective components to yield a resultant vector that represents both magnitude and direction.

Consider two vectors, \( \boldsymbol{A} = (A_x, A_y, A_z) \) and \( \boldsymbol{B} = (B_x, B_y, B_z) \). When you perform vector addition, you simply add the corresponding components of each vector:
  • \( A_x + B_x \)
  • \( A_y + B_y \)
  • \( A_z + B_z \)
The result is a new vector, \( \boldsymbol{C} = (A_x + B_x, A_y + B_y, A_z + B_z) \).

An important property of vector addition is that it is commutative. This means that the order in which you add the vectors does not matter. For example, \( \boldsymbol{A} + \boldsymbol{B} \) will yield the same result as \( \boldsymbol{B} + \boldsymbol{A} \). This property arises from the basic arithmetic rule that addition of numbers is commutative.
Scalar Product
The scalar product, also known as the dot product, is a way to multiply vectors that results in a scalar, or a single number. It is useful in determining the angle between two vectors and in projections.

Given two vectors, \( \boldsymbol{A} = (A_x, A_y, A_z) \) and \( \boldsymbol{B} = (B_x, B_y, B_z) \), the scalar product is calculated as follows:
  • \( \boldsymbol{A} \cdot \boldsymbol{B} = A_xB_x + A_yB_y + A_zB_z \)
Each term in the sum represents the product of the components of the vectors along each coordinate axis.

A key characteristic of the scalar product is its commutative nature. This means that swapping the vectors in the multiplication does not change the result: \( \boldsymbol{A} \cdot \boldsymbol{B} = \boldsymbol{B} \cdot \boldsymbol{A} \). This commutativity is due to the intrinsic properties of the multiplication of real numbers, reflecting how each paired multiplication within the product is unchanged by the order.
Vector Product
The vector product, or cross product, is a way to multiply two vectors resulting in another vector. This outcome is crucial when dealing with rotational themes in physics, as the resulting vector is perpendicular to the plane formed by the original vectors.

For vectors \( \boldsymbol{A} = (A_x, A_y, A_z) \) and \( \boldsymbol{B} = (B_x, B_y, B_z) \), the vector product is represented as:
  • \( \boldsymbol{A} \times \boldsymbol{B} = (A_yB_z - A_zB_y, A_zB_x - A_xB_z, A_xB_y - A_yB_x) \)
Each component of the resulting vector is a determinant made up of components from \( \boldsymbol{A} \) and \( \boldsymbol{B} \).

Unlike the previous concepts, the vector product is not commutative; instead, it is anticommutative. This means reversing the order of the vectors changes the sign of the resultant vector: \( \boldsymbol{A} \times \boldsymbol{B} = - (\boldsymbol{B} \times \boldsymbol{A}) \). This anticommutative nature is linked to the geometrical interpretation of the cross product, where the direction of the resulting vector depends on the order of the original vectors.

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Most popular questions from this chapter

Later in our sudy of physics we will encounter quantities represented by \((\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\) , (a) Prove that for any three vectors \(\vec{A}, \vec{B},\) and \(\overrightarrow{\boldsymbol{C}}, \overrightarrow{\boldsymbol{A}} \cdot(\overrightarrow{\boldsymbol{B}} \times \overrightarrow{\boldsymbol{C}})=(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}) \cdot \overrightarrow{\boldsymbol{C}}\) (b) Calculate \((\vec{A} \times \vec{B}) \cdot \vec{C}\) for the three vectors \(\vec{A}\) with magnitude \(A=5.00\) and angle \(\theta_{A}=26.0^{\circ}\) measured in the sense from the \(+x\)-axis toward the \(+y\) -axis, \(\overrightarrow{\boldsymbol{B}}\) with \(B=4.00\) and \(\theta_{B}=63.0^{\circ},\) and \(\overrightarrow{\boldsymbol{C}}\) with magnitude 6.00 and in the \(+z\) -direction. Vectors \(\overrightarrow{\boldsymbol{A}}\) and \(\overrightarrow{\boldsymbol{B}}\) are in the \(x y\) -plane.

The following conversions occur frequently in physics and are very useful. (a) Use 1 mi \(=5280 \mathrm{ft}\) and \(1 \mathrm{h}=3600 \mathrm{s}\) to convert 60 \(\mathrm{mph}\) to units of \(\mathrm{ff} / \mathrm{s} .(\mathrm{b})\) The acceleration of a freely falling object is 32 \(\mathrm{ff} / \mathrm{s}^{2} .\) Use \(1 \mathrm{ft}=30.48 \mathrm{cm}\) to express this acceleration in units of \(\mathrm{m} / \mathrm{s}^{2} .\) (c) The density of water is 1.0 \(\mathrm{g} / \mathrm{cm}^{3} .\) Convert this density to units of \(\mathrm{kg} / \mathrm{m}^{3} .\)

The two vectors \(\overrightarrow{\boldsymbol{A}}\) and \(\overrightarrow{\boldsymbol{B}}\) are drawn from a common point, and \(\overrightarrow{\boldsymbol{C}}=\overrightarrow{\boldsymbol{A}}+\overrightarrow{\boldsymbol{B}},\) (a) Show that if \(\boldsymbol{C}^{2}=\boldsymbol{A}^{2}+\boldsymbol{B}^{2},\) the angle between the vectors \(\overrightarrow{\boldsymbol{A}}\) and \(\overrightarrow{\boldsymbol{B}}\) is \(90^{\circ} .\) (b) Show that if \(C^{2}< A^{2}+B^{2},\) the angle between the vectors \(\vec{A}\) and \(\vec{B}\) is greater than \(90^{\circ}\) (c) Show that if \(C^{2}>A^{2}+B^{2},\) the angle between the vectors \(\vec{A}\) and \(\vec{B}\) is between \(0^{\circ}\) and \(90^{\circ} .\)

As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of \(8.50 \pm 0.02 \mathrm{cm}\) and a thickness of \(0.050 \pm 0.005 \mathrm{cm} .\) (a) Find the average volume of a cookie and the uncertainty in the volume. (b) Find the ratio of the diameter to the thickness and the uncertainty in this ratio.

A useful and easy-to-remember approximate value for the number of seconds in a year is \(\pi \times 10^{7} .\) Determine the percent error in this approximate value. (There are 365.24 days in one year.)
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