/*! 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 11 Use the dot product to find the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 11

Use the dot product to find the angle between the vectors (2,3) and (3,4) .

Short Answer

Expert verified
The angle between the vectors (2,3) and (3,4) is approximately \(26.4^{\circ}\).

Step by step solution

01

Find the dot product

Calculate the dot product of the two given vectors (2,3) and (3,4) using the formula \((a_1b_1+a_2b_2)\), where \((a_1,a_2)\) and \((b_1,b_2)\) are the components of the vectors: \[\text{Dot product} = (2 \times 3) + (3 \times 4)\]
02

Calculate the dot product

Use the formula from Step 1 to find the dot product of vectors (2,3) and (3,4): \[\text{Dot product} = (2 \times 3) + (3 \times 4) = 6+12 =18\]
03

Find the magnitudes of each vector

Calculate the magnitude of each vector, (2,3) and (3,4), using the formula \(||\vec{a}|| = \sqrt{a_1^2+a_2^2}\): \[||\vec{a}|| = \sqrt{2^2+3^2} = \sqrt{4+9} = \sqrt{13}\] \[||\vec{b}|| = \sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5\]
04

Use the dot product formula to find the cosine of the angle

Use the dot product formula to find the cosine of the angle between the two vectors: \[\cos(\theta) = \frac{\text{Dot product}}{||\vec{a}|| \times ||\vec{b}||} = \frac{18}{\sqrt{13} \times 5}\]
05

Find the angle

To find the angle between the two vectors, apply the arccosine function to the cosine value calculated in Step 4: \[\theta = \arccos(\frac{18}{\sqrt{13} \times 5})\] Now, use a calculator to find the approximate value of the angle in degrees: \[\theta \approx 26.4^{\circ}\] Therefore, the angle between the vectors (2,3) and (3,4) is approximately 26.4 degrees.

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!

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

Sketch the graph of the function \(\sin ^{2} x\) on the interval \([-3 \pi, 3 \pi]\).

P Suppose \(-\frac{\pi}{2}<\theta<0\) and \(\cos \theta=0.8\) (a) Without using a double-angle formula, evaluate \(\cos (2 \theta)\) (b) Without using an inverse trigonometric function, evaluate \(\cos (2 \theta)\) again.

Sketch the graph of the function \(\cos ^{2}(3 x)\) on the interval \([-2 \pi, 2 \pi]\).

Find the center and radius of the circle whose equation in polar coordinates is \(r=3 \cos \theta\).

Convert the rectangular coordinates given for each point to polar coordinates \(r\) and \(\theta .\) Use radians, and always choose the angle to be in the interval \((-\pi, \pi)\). $$ (3,2) $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

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.