/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Find each of the following dot p... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 8

Find each of the following dot products. $$ \langle 10,8\rangle \cdot\langle 12,-6\rangle $$

Short Answer

Expert verified
The dot product is 72.

Step by step solution

01

Understand the Dot Product Formula

The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is calculated using the formula \( a \cdot c + b \cdot d \). In this exercise, we have the vectors \( \langle 10, 8 \rangle \) and \( \langle 12, -6 \rangle \).
02

Multiply Corresponding Components

Calculate the product of the first components: \( 10 \times 12 = 120 \). Next, calculate the product of the second components: \( 8 \times (-6) = -48 \).
03

Sum the Results

Add the results from the previous step: \( 120 + (-48) = 72 \).
04

State the Final Answer

The dot product of \( \langle 10, 8 \rangle \) and \( \langle 12, -6 \rangle \) is \( 72 \).

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 91影视!

Key Concepts

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

Vector Multiplication
When we talk about vector multiplication, we usually mean the dot product or cross product. In this article, we focus on the dot product, a method of multiplying two vectors to get a single scalar (a regular number).
The dot product tells us how much one vector goes in the same direction as another. To find this, we multiply the corresponding components of the vectors and then add these products together.
  • Formula: For vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \), the dot product is \( a \cdot c + b \cdot d \).
  • Result: The outcome is a single number, not a vector.
Understanding vector multiplication through the dot product can help in various applications such as physics, engineering, and computer graphics.
Vector Components
Vectors are often represented in component form, like \( \langle a, b \rangle \). Each number inside is a component, representing how far the vector stretches in each direction.
This helps us break down complex directions into manageable parts. Components are crucial because they allow us to perform arithmetic, such as adding, subtracting, and finding dot products between vectors easily.
  • First component: Represents the horizontal direction (x-axis).
  • Second component: Represents the vertical direction (y-axis).
This breakdown into components simplifies many operations, making calculations straightforward and providing deeper insight into vector behavior.
Vector Operations
Vectors can undergo several operations, like addition, subtraction, and multiplication. Each operation has specific rules and gives different results.
In this guide, our focus is on the dot product, a type of vector multiplication. Here are some basics of vector operations:
  • Addition: Combine vectors by adding their corresponding components.
  • Subtraction: Find the difference by subtracting corresponding components.
  • Dot product: As explained, multiply and sum components.
Each operation has its use, applicable in fields ranging from physics to data science, allowing us to model and solve real-world problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A force of 1000 pounds is acting on an object at an angle of \(45^{\circ}\) from the horizontal. Another force of 500 pounds is acting at an angle of \(-40^{\circ}\) from the horizontal. What is the direction angle of the resultant force?

In Exercises 95-98, determine whether each statement is true or false. Equal vectors must coincide.

A baseball pitcher throws a ball with an initial velocity of 100 feet per second at an angle of \(5^{\circ}\) with the horizontal. What are the vertical and horizontal components of the velocity?

In Exercises 69-72, given \(u=\langle a, b\rangle\) and \(v=\langle c, d\rangle\), show that the following properties are true. $$ \mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|^{2} $$

For Exercises 67 and 68, refer to the following to find the dot product: The dot product of vectors with \(n\) component is $$ \left\langle a_{1}, a_{2}, \ldots, a_{n}\right\rangle \cdot\left\langle b_{1}, b_{2}, \ldots, b_{n}\right\rangle=a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n} $$ $$ \langle 1,0,-2,3\rangle \cdot\langle 5,2,3,1\rangle $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.