91Ó°ÊÓ

We study the dot product of two vectors. Given two vectors \(\mathbf{A}=\left\langle x_{1}, y_{1}\right\rangle\) and \(\mathbf{B}=\left\langle x_{2}, y_{2}\right\rangle,\) we define the dot product \(\mathbf{A} \cdot \mathbf{B}\) as follows: $$\mathbf{A} \cdot \mathbf{B}=x_{1} x_{2}+y_{1} y_{2}$$ For example, if \(\mathbf{A}=\langle 3,4\rangle\) and \(\mathbf{B}=\langle-2,5\rangle,\) then \(\mathbf{A} \cdot \mathbf{B}=(3)(-2)+(4)(5)=14 .\) Notice that the dot product of two vectors is a real number. For this reason, the dot product is also known as the scalar product. For Exercises \(61-63\) the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are defined as follows: $$\mathbf{u}=\langle-4,5\rangle \quad \mathbf{v}=\langle 3,4\rangle \quad \mathbf{w}=\langle 2,-5\rangle$$ (a) Compute \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{v} \cdot \mathbf{u}\) (b) Compute \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{w} \cdot \mathbf{v}\) (c) Show that for any two vectors \(\mathbf{A}\) and \(\mathbf{B}\), we have \(\mathbf{A} \cdot \mathbf{B}=\mathbf{B} \cdot \mathbf{A} .\) That is, show that the dot product is commutative. Hint: Let \(\mathbf{A}=\left\langle x_{1}, y_{1}\right\rangle,\) and let \(\mathbf{B}=\left\langle x_{2}, y_{2}\right\rangle\)

Step by step solution

01

Calculate \( \mathbf{u} \cdot \mathbf{v} \)

Given \( \mathbf{u} = \langle -4, 5 \rangle \) and \( \mathbf{v} = \langle 3, 4 \rangle \), apply the dot product formula: \( \mathbf{u} \cdot \mathbf{v} = (-4)(3) + (5)(4) = -12 + 20 = 8 \).

02

Calculate \( \mathbf{v} \cdot \mathbf{u} \)

Apply the dot product formula to vectors \( \mathbf{v} = \langle 3, 4 \rangle \) and \( \mathbf{u} = \langle -4, 5 \rangle \). So, \( \mathbf{v} \cdot \mathbf{u} = (3)(-4) + (4)(5) = -12 + 20 = 8 \).

03

Calculate \( \mathbf{v} \cdot \mathbf{w} \)

Given \( \mathbf{v} = \langle 3, 4 \rangle \) and \( \mathbf{w} = \langle 2, -5 \rangle \), use the formula: \( \mathbf{v} \cdot \mathbf{w} = (3)(2) + (4)(-5) = 6 - 20 = -14 \).

04

Calculate \( \mathbf{w} \cdot \mathbf{v} \)

Apply the dot product formula with vectors \( \mathbf{w} = \langle 2, -5 \rangle \) and \( \mathbf{v} = \langle 3, 4 \rangle \). Thus, \( \mathbf{w} \cdot \mathbf{v} = (2)(3) + (-5)(4) = 6 - 20 = -14 \).

05

Prove Commutativity of the Dot Product

The dot product of two vectors \( \mathbf{A} = \langle x_1, y_1 \rangle \) and \( \mathbf{B} = \langle x_2, y_2 \rangle \) is defined as \( \mathbf{A} \cdot \mathbf{B} = x_1 x_2 + y_1 y_2 \). Meanwhile, \( \mathbf{B} \cdot \mathbf{A} = x_2 x_1 + y_2 y_1 \). Both expressions are the same (since multiplication is commutative), concluding that \( \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} \). Therefore, the dot product is commutative.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Vectors

Vectors are mathematical entities that represent both magnitude and direction. They are often depicted as arrows in space, where the length of the arrow indicates the magnitude, and the arrowhead points in the direction. Vectors are commonly expressed in component form, such as \( \mathbf{A} = \langle x, y \rangle \) in two dimensions. Each vector consists of components along the coordinate axes. For instance, in the vector \( \langle 3, 4 \rangle \), the numbers 3 and 4 are the components along the x and y axes, respectively.

Vectors can be used to represent quantities such as displacement, velocity, and force. In physics and engineering, they enable the representation of real-world phenomena by considering both how large something is and in which direction it moves or applies. Working with vectors involves operations such as addition, subtraction, and multiplication. These operations allow various computations and transformations in graphical space.

Scalar Product

The scalar product, also known as the dot product, is a method of multiplying two vectors to yield a scalar, a real number, rather than another vector. It is denoted as \( \mathbf{A} \cdot \mathbf{B} \) for vectors \( \mathbf{A} \) and \( \mathbf{B} \). The formula for the scalar product in two dimensions is \( \mathbf{A} \cdot \mathbf{B} = x_1 x_2 + y_1 y_2 \), where \( x_1, y_1 \) are the components of \( \mathbf{A} \), and \( x_2, y_2 \) are the components of \( \mathbf{B} \).

The scalar product has important applications:

• It determines how much of one vector goes in the direction of another.
• It finds the angle between vectors, using the relationship \( \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \).
• It is used to project vectors onto each other.

The scalar product is crucial in various fields, including physics, computer graphics, and engineering, where it aids in analyzing projections, light, and forces.

Commutativity

Commutativity is a fundamental property in mathematics, which states that the order in which two numbers or operations are combined does not affect the result. For the dot product, this means that \( \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A} \). This property is easily demonstrated by observing the definition of the dot product:

For vectors \( \mathbf{A} = \langle x_1, y_1 \rangle \) and \( \mathbf{B} = \langle x_2, y_2 \rangle \), we have:

• \( \mathbf{A} \cdot \mathbf{B} = x_1 x_2 + y_1 y_2 \)
• \( \mathbf{B} \cdot \mathbf{A} = x_2 x_1 + y_2 y_1 \)

Due to the commutative nature of multiplication, both expressions are identical, proving that the dot product is commutative.

Understanding the commutativity of the dot product helps simplify calculations and ensures that reversing the order of vectors in a dot product operation will not impact the outcome.

Precalculus

Precalculus serves as the stepping stone to more advanced mathematical concepts, preparing students for calculus by reinforcing algebra and introducing new topics like trigonometry, complex numbers, and vectors.

One area of study in precalculus is vectors and vector operations. Understanding vectors entails mastering their representation, operations such as vector addition and scalar multiplication, and importantly, the dot product. These foundational skills are essential for tackling calculus concepts like multivariable functions and vector fields.

Another important topic covered in precalculus is the exploration of functions and their properties, which require a solid understanding of mathematical reasoning and analysis. By the time students complete precalculus, they should be equipped with the tools necessary to understand higher-order mathematics, enabling a smoother transition into calculus and related fields like physics and engineering.