×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

39

0

4

# Introduction to Vectors Part 1 MATH 241

Marketplace > University of Maryland - College Park > Math > MATH 241 > Introduction to Vectors Part 1

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

This covers the basics of the three dimensional coordinate system and vectors while also providing examples.
COURSE
Calculus III
PROF.
Dr. Roohollah Ebrahimian
TYPE
Class Notes
PAGES
4
WORDS
CONCEPTS
Math, Calculus, Multivariable, vectors, coordinates, magnitude
KARMA
25 ?

## Popular in Math

This 4 page Class Notes was uploaded by Saadiq Shaik on Saturday September 10, 2016. The Class Notes belongs to MATH 241 at University of Maryland - College Park taught by Dr. Roohollah Ebrahimian in Fall 2016. Since its upload, it has received 39 views. For similar materials see Calculus III in Math at University of Maryland - College Park.

×

## Reviews for Introduction to Vectors Part 1

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/10/16
Introduction to Vectors Part 1 Saadiq Shaik September 2016 1 Cartesian Coordinates in Space A vector is de▯ned as a quantity with both a magnitude and direction. We will be dealing with vectors in three dimensional space, so familiarize yourself with the three dimensional coordinate system. The coordinate planes are the xy-plane, yz-plane, and xz-plane. There are eight octants, but the one we will be dealing with most is the ▯rst octant, which contains positive x, y, and z values. Imagine a point P with the coordinates (a ;b ;c ) and a point Q with 1 1 1 the coordinates 2a2;b3;c ) in three dimensional space. The distance between them is given by the distance formula, q jPQj = (1 ▯ a2) + (1 ▯ b2) + (1 ▯ 2 ) Example 1 p p p Find the distance between the points P = (▯1;0;2 and Q = (1; 2; 3). Solution Simple substitution gives us the answer. q p p p jPQj = (▯1 ▯ 1) + (0 ▯ 2) + (2 3 ▯ 3) p jPQj = 4 + 2 + 3 p jPQj = 9 = 3 1 Likewise, the midpoint of P and Q is given by the midpoint formula, ▯a1+ a 2 b1+ b 2 c1+ c 2▯ M = ; ; 2 2 2 Example 2 1 1 Find the midpoint of points Q = ( ;2;0) and R = ( ;322): Solution Substituting the values in gives us ▯ 1+ 1 1 + 3 0 + 2 ▯ M = 2 2; ; 2 2 2 ▯ 1 4 2 ▯ M = ; ; 2 2 2 1 M = ( 22;1) A sphere of radius r centered at (x0;y0;z0) is de▯ned by the equation of a sphere in space where (x;y;z) is the set of all points on the sphere r = (x ▯ x )0+ (y ▯ y ) 0 (z ▯ z ) 0 2 Example 3 Show that x + y + z = 4x + 4y ▯ 6z is an equation of a sphere. Find its center and radius. Solution By moving the terms to the right and completing the square we get 2 2 2 (x ▯ 4x + 4) + (y ▯ 4y + 4) + (z + 6z + 9) = 4 + 4 + 9 or (x ▯ 2) + (y ▯ 2) + (z + 3) = 17 p This is an equation of the sphere with center (2, 2, -3) and 17. 2 2 Vectors in Space Given the points P = (a ;b ;c ) and Q = (a ;b ;c ), the vector PQ is de▯ned 0 0 0 1 1 1 as (a1▯ a 0b 1 b ;0 ▯1c ) 0 and can be rewritten with the unit vectors i = (1;0;0), j = (0;1;0), and k = (0;0;1) as (a1▯ a 0i + (b 1 b 0j + (c 1 c )0 or ai + bj + ck where a, b, and c are the di▯erences of the x, y, and z coordinates respec- tively. They are called the vector components. Example 4 ▯! Given P = (1;2;3) and Q = (0;2;4), write the vector PQ in unit vector notation. Solution We start by ▯nding the coe▯cients a, b and c, a = 0 ▯ 1 = ▯1;b = 2 ▯ 2 = 0;c = 4 ▯ 3 = 1 and then plug them into the formula to obtain ▯1i + 0j + 1k or ▯i + k The magnitude of a vector a is given by the magnitude formula q jjajj = a1+ a 2 a 3 This can also be called the length or norm of the vector. 3 Example 5 Find the magnitude of the vector a = 2i + 3j +3k. Solution We simply ▯nd the vector components and substitute them to arrive at q 2 2 p 2 jjajj = 2 + 3 + 3 or p jjajj = 4 + 9 + 3 = 16 = 4 A vector in a plane, where3a = 0, can be rewritten as a = jjajj(cos(▯i) + sin(▯j)) where ▯ is the angle between the x-axis and a. Here is an important point regarding vectors: jjcajj = jcjjjajj where c is some constant. If b = jcjjjajj, then the two vectors are parallel. The unit vector in the direction of a vector a is given by a jjajj Vectors can added by adding their components and subtracted by subtracting their components, as well. 4

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Steve Martinelli UC Los Angeles

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Allison Fischer University of Alabama

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over \$600 per month. I LOVE StudySoup!"

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com