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Pre-Calculus

Vectors

www.njctl.org 2015-03-24

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Table of Contents

Intro to Vectors Operations with Vectors Addition Subtraction Dot Product Scalar Multiples Angle Between Vectors 3-Dimensional Space Vectors, Lines, and Planes Vector Equations of Lines Converting Rectangular and Polar Forms

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Intro to Vectors

Return to Table of Contents

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Drawing A Vector

Remember displacement is the distance away from your initial position, it does not account for the actual distance you moved

A vector is always drawn with an arrow at the tip indicating the direction, and the length of the line determines the magnitude.

Intro to Vectors

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Determining magnitude and direction

anti-parallel

All of these vectors have the same magnitude, but vector B runs anti-parallel therefore it is denoted negative A.

Intro to Vectors

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Draw a vector to represent: a plane flying North-East at 500 mph a wind blowing to the west at 50 mph

Intro to Vectors

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Draw a vector to represent: a boat traveling west at 4 knots a current traveling south at 2 knots

Intro to Vectors

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1 Which vector represents a car driving east at 60 mph?

A

B C D

Intro to Vectors

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2 Which vector represents a wind blowing south at 30 mph if the given vector represents a car driving east at 60mph?

A

B C D

Intro to Vectors

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3 A rabbit runs east 30 feet and then north 40 feet, which vector represents the rabbits displacement from its starting position?

A

B C D

Intro to Vectors

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There are 2 kinds of vectors drawn on a coordinate grid: Those that start from the origin: Those that don't start at the origin: joins (2 , -4) to (4 , 4)

Intro to Vectors

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A vector can be broken down into its component forces. ux is the horizontal part. uy is the vertical part. In ux = 8 uy = 6

Intro to Vectors

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joins (2 , -4) to (4 , 4)

Intro to Vectors

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4 Which vector is in standard position?

A

B C D

Intro to Vectors

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5 What is ?

Intro to Vectors

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6 What is ?

Intro to Vectors

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7 What is ?

Intro to Vectors

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8 What is ?

Intro to Vectors

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9 What is ?

Intro to Vectors

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10 What is ?

Intro to Vectors

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11 What is ?

Intro to Vectors

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12 What is ?

Intro to Vectors

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After a vector is broken down into its component forces,the magnitude of the vector can be calculated. ux is the horizontal part. uy is the vertical part. In ux = 8 uy = 6 8 6 = magnitude (length) of

Intro to Vectors

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A plane is traveling with an eastern component of 40 miles and a northern component of 75 miles, every hour. Draw a representation of this, include the vector of the planes actual path. What is the planes displacement traveled after one hour?

Intro to Vectors

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13 If a car under goes a displacement of 3 km North and another of 4 km to the East what is the car's displacement?

A 5#2

B 7 C 5 D 4

E 3

3 km 4 km x

Intro to Vectors

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14 If a car under goes a

displacement of 7 km North and another of 3 km to the West what is the car's displacement?

A

B C D

E

3 km 7 km x

Intro to Vectors

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15 If a car under goes a

displacement of 12 km South and another of 5 km to the East what is the car's displacement?

A

B C D

E

Intro to Vectors

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16 What is the magnitude of ?

Intro to Vectors

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17 What is the magnitude of ?

Intro to Vectors

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18 What is the magnitude of that connects (2,1) to (5,10)?

Intro to Vectors

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19 What is the magnitude of that connects (-4,3) to (-5,-3)?

Intro to Vectors

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20 The components of vector A are given as follows:

A 2.9

B 6.9 C 9.3 D 18.9

E 47.5 The magnitude of A is closest to:

Intro to Vectors

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21 The components of vector A are given as follows:

A 4.2

B 8.4 C 11.8 D 18.9

E 70.9 The magnitude of A is closest to:

Intro to Vectors

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A plane is traveling with an eastern component of 40 miles and a northern component of 75 miles, every hour. Draw a representation of this, include the vector of the planes actual path. How many degrees North of East is the plane traveling? This is called the vectors direction.

Intro to Vectors

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22 What is the direction of that connects (2,1) to (5,10)?

Intro to Vectors

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23 What is the direction of that connects (-4,3) to (-5,-3)?

Intro to Vectors

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24 The components of vector A are given as follows:

A 52.7

B

55.3

C

62.3

D

297.7 E 307.3 The direction of A,measured counterclockwise from the east, is closest to:

Intro to Vectors

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25 The components of vector A are given as follows:

A 37.3

B

52.7

C

139.6

D

307.3

E

322.7

The direction of A,measured counterclockwise from the east, is closest to:

Intro to Vectors

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Converting Rectangular and Polar Forms

Return to Table of Contents

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A vector in standard position can be written in 2 ways: When a vector is given as (x,y) it is in rectangular form. When a vector is given (r,#) it is in polar form.

Converting Rectangular and Polar Forms

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To Convert from Rectangular to Polar

(-6,8)

Converting Rectangular and Polar Forms

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26 Vector A is in standard position and in rectangular form (2,7). What is its magnitude (ie radius)?

Converting Rectangular and Polar Forms

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27 Vector A is in standard position and in rectangular form (2,7). What is its direction?

Converting Rectangular and Polar Forms

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28 Vector A is in standard position and in rectangular form (-3,8). What is its magnitude (ie radius)?

Converting Rectangular and Polar Forms

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29 Vector A is in standard position and in rectangular form (-3,8). What is its direction?

Converting Rectangular and Polar Forms

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30 Vector A is in standard position and in polar form (3,70). What is Ax?

Converting Rectangular and Polar Forms

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31 Vector A is in standard position and in polar form (3,70). What is Ay?

Converting Rectangular and Polar Forms

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32 Vector A is in standard position and in polar form (3,110). What is A

x?

Converting Rectangular and Polar Forms

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33 Vector A is in standard position and in polar form (3,110). What is Ay?

Converting Rectangular and Polar Forms

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Operations with Vectors

Return to Table of Contents

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Scalar Multiples

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Scalar versus Vector

A scalar has only a physical quantity such as mass, speed, and time. A vector has both a magnitude and a direction associated with it, such as velocity and acceleration. A vector is denoted by an arrow above the variable,

Scalar Multiples

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A car travels East at 50 mph, how far does it go in 3 hours? East at 50 mph is represented by a vector 3 hours is a scalar multiplier, which is why there are 3 vectors. The car traveled 150 miles in 3 hours.

Scalar Multiples

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34 What is speed?

A Vector

B

Scalar

Scalar Multiples

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35 Which diagram represents a person walking Northeast at 4 mph for 2 hours?

A

B C D

Scalar Multiples

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When vectors are on the coordinate plane a scalar can be used on the ordered pair(s). joins (2 , -4) to (4 , 4) joins (4, -8) to (8 , 8) What happened to the magnitude? What happen to the direction?

Scalar Multiples

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Given and Find:

Scalar Multiples

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36 Given vector =(4, 5) what is

A

(8 , 10)

B

(6 , 7)

C

(16 , 25)

D

(8, 7)

Scalar Multiples

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37 Given vector =(4, 5) what is

A

(8/3 , 10/3)

B

(2 , 7/3)

C

(16/3 , 25/3)

D

(8/3, 7/3)

Scalar Multiples

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38 Given vector =(4, 5) what is

A

(1 , 2)

B

(12 , 15)

C

(-12 , 15)

D

(-12, -15)

Scalar Multiples

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39 Given find

Scalar Multiples

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Addition

Return to Table of Contents

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Vector Addition

Addition

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Vector Addition Methods

Tail to Tip Method

Addition

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Vector Addition Methods

Move the vectors to represent the following operation. Draw the resultant vector.

Addition

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Vector Addition Methods

Parallelogram Method Place the tails of each vector against one

• another. If you finish

drawing the parallelogram with dashed lines and draw a diagonal line from the tails to the other end of the parallelogram to find the vector sum.

Addition

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Given Find the resultant vector:

Addition

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40 What is the resultant vector for if

A

(5 , 5)

B

(4 , 6)

C

(5 , 6)

D

(4 , 5)

Addition

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41 What is if

Addition

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42 The components of vectors

A 5

B #17 C 17 D 10

E 8 and are given as follows: Solve for the magnitude of

Addition

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Subtraction

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Anti- Parallel Vectors

Subtraction

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Subtraction

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Vector Addition Method for Subtraction

Draw the vectors to represent the following operation. Draw the resultant vector.

Subtraction

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Vector Addition Method for Subtraction

Draw the vectors to represent the following operation. Draw the resultant vector.

Subtraction

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Vector Addition Method for Subtraction

Draw the vectors to represent the following operation. Draw the resultant vector.

Subtraction

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Vector Addition Method for Subtraction

Draw the vectors to represent the following operation. Draw the resultant vector.

Subtraction

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43

A

B

C D

Subtraction

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44

A

B

C D

Subtraction

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45

A

B C D

Subtraction

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Given Find the resultant vector:

Subtraction

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46 What is the resultant vector for if

A

(3 , 1)

B

(-3 , -1)

C

(5 , 5)

D

(3 , -1)

Subtraction

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47 What is if

Subtraction

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48 The components of vector A and B are given as follows:

A 10.17

B 4.92 C 2.8 D 9.7

E 25 The magnitude of B-A, is closest to:

Subtraction

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Vector Equations

• f Lines

Return to Table of Contents

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Vector Equation for a Line

R S

Consider the line through R and S. There is a unique congruent vector, in standard position. The difference between any two points on the line is where t is a real number.

Vector Equations of Lines

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Vector Equation for a Line

R S Example: Find the equation of the line through R(2,6) and is parallel to v.

Vector Equations of Lines

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Draw a graph of the line through (2, 7) and parallel to v=(1,4) Write the equation of the line. Write the equation of the line in parametric form.

Vector Equations of Lines

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Draw a graph of the line through (3, -7) and parallel to v=(-2,6) Write the equation of the line. Write the equation of the line in parametric form.

Vector Equations of Lines

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49 Which of the following is the vector equation of the line through (-3, -4) and parallel to u=(6, 1)?

A

(x+4, y+3) = t(6,1)

B

(x-4, y-3) = t(6,1)

C

(x+3, y+4) = t(6,1)

D

(x-3, y-4) = t(6,1)

Vector Equations of Lines

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50 Which of the following is parametric form of the equation

• f the line through (-3, -4) and parallel to u=(6, 1)?

A

B C D

Vector Equations of Lines

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51 Which of the following is the vector equation of the line through (5, 2) and parallel to u=( -7, 1)?

A

(x -1, y+7) = t(5,2)

B

(x+7, y-1) = t(5,2)

C

(x+5, y+2) = t(-7,1)

D

(x-5, y-2) = t(-7,1)

Vector Equations of Lines

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52 Which of the following is parametric form of the equation

• f the line through (5, 2) and parallel to u=( -7, 1)?

A

B C D

Vector Equations of Lines

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Find the parametric equation of the line through (4, 7) and (2, 8)

Vector Equations of Lines

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Given two points (x1,y1) and (x2,y2), the parametric equations of the line are: x = x1 + t*(x2 - x1) y = y1 + t*(y2 - y1)

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Find the parametric equation of the line through (3, -5) and (8, 9)

Vector Equations of Lines

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53 Which of the following is a parametric form of the equation

• f the line through ( -2, 0) and (4, 7)?

A

B C D

Vector Equations of Lines

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54 Which of the following is a parametric form of the equation

• f the line through (4, 9) and (8,3)?

A

B C D

Vector Equations of Lines

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Find the vector equation of:

Vector Equations of Lines

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Dot Product

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The dot product, also called a scalar product, returns a single numerical value between 2 vectors. Example: Given: Find:

Dot Product

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Given Find the resultant vector:

Dot Product

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56Given , find

Dot Product

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57Given , find

Dot Product

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58Given , find

Dot Product

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Vectors are perpendicular/orthogonal/normal, if their dot product is zero.

If and Are the vectors perpendicular? Justify your answer.

Dot Product

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Vectors form an

• btuse angle if

dot product < 0 Vectors form an acute angle if dot product > 0

If and What kind of angle do the vectors form? Justify your answer.

Dot Product

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59Are these vectors perpendicular?

Yes No

Dot Product

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60Are these vectors orthogonal?

Yes No

Dot Product

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61Are these vectors normal?

Yes No

Dot Product

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62What kind of angle is formed by the following vectors?

A

acute

B

right

C

• btuse

Dot Product

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63What kind of angle is formed by the following vectors?

A

acute

B

right

C

• btuse

Dot Product

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64What kind of angle is formed by the following vectors?

A

acute

B

right

C

• btuse

Dot Product

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Angle Between Vectors

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Angle Between Vectors is the dot product of the vectors is the product of the magnitudes

Angle Between Vectors

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Find the angle between and

Angle Between Vectors

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Recall that during the study of dot product that an angle was obtuse if its dot product was negative. cos-1 returns an obtuse value only if And the only way for is if

Angle Between Vectors

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65Find the angle between and

Angle Between Vectors

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66Find the angle between and

Angle Between Vectors

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67Find the angle between and

Angle Between Vectors

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68A cruise ship is being towed by 2 tug boats. The first is pulling the ship 100m east and 20m north. The second is pulling the ship 40m east and 75m north. What is the angle between the tow lines?

Angle Between Vectors

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69A cruise ship is being towed by 2 tug boats. The first is pulling the ship 100m east and 20m north. The second is pulling the ship 40m east and 75m north. What is the angle the ship is going relative to east?

Angle Between Vectors

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3-Dimensional Space

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y z x (-) (-) (-) (+) (+) (+)

3-Dimensional Space

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y z x (-) (-) (-) (+) (+) (+) Graph (2,3,4)

3-Dimensional Space

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y z x (-) (-) (-) (+) (+) (+) Graph (-2, 3, -4)

3-Dimensional Space

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y z x (-) (-) (-) (+) (+) (+) Graph (0,3,5)

3-Dimensional Space

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70What are the coordinates of the point?

A

(1,3,6)

B

(1,6,3)

C

(3,1,6)

D

(3,6,1)

1 3 6

3-Dimensional Space

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71What are the coordinates of the point?

A

(5,0,2)

B

(0,2,5)

C

(2,0,5)

D

(2,5,0)

2 5

3-Dimensional Space

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72What is the distance between (2,3,4) and (-2,-3,-4)?

3-Dimensional Space

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73What is the distance between (3,0,7) and (-6,-2,5)?

3-Dimensional Space

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74What is the length of a diagonal ofa box with dimesions 4 x 6 x 7?

3-Dimensional Space

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Vectors, Lines, and Planes

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Vector Operations in 3-Space Addition: Subtraction: Scalar Multiplication: Dot Product: Angle Between Vectors: Orthogonal Vectors: Dot product =0

Vectors, Lines, & Planes

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Find: What is the angle between and

Vectors, Lines, & Planes

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Lines in Space Point P = (2, 6, -3) is on line m and parallel to v = (4, -4, 2). Write the equation of m. Vector Equation Parametric

Vectors, Lines, & Planes

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Equation of a Plane

3x +4y +2z =12

4 3 6

The intersection of the plane and the xy-plane is

• line. So although the

figure looks like a triangle it continues forever.

Vectors, Lines, & Planes

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Equation of a Plane

3x +4y +2z =12

4 3 6

The vector v=(3, 4, 2) is perpendicular to the plane.

Vectors, Lines, & Planes

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Example: Write the equation of the line perpendicular to 3x +8y - 12z =24 passing through P=(1, -5, 7).

Vectors, Lines, & Planes

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Example: Write the equation of plane passing through (1, 3, 5) and perpendicular to v = (-2, 3, 4).

Vectors, Lines, & Planes

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79What is the x-intercept of 3x + 8y + 12z = 24?

Vectors, Lines, & Planes

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80What is the y-intercept of 3x + 8y + 12z = 24?

Vectors, Lines, & Planes

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81What is the z-intercept of 3x + 8y + 12z = 24?

Vectors, Lines, & Planes

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82Which is the equation of the line perpendicular to 3x +8y +12z = 24 and passing through (2, -4, 6)?

A

(x+2, y-4, z+6)=t(3, 8, 12)

B

(x-2, y+4, z-6)=t(3, 8, 12)

C

(x+2, y+4, z+6)=t(3, 8, 12)

D

(x-2, y-4, z-6)=t(3, 8, 12)

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83Which is the equation of the line perpendicular to x -2y +7z = 14 and passing through (-1, 4, -8)?

A

(x+1, y-4, z+8)=t(1, -2, 7)

B

(x+1, y-4, z+8)=t(-1, 2, -7)

C

(x-1, y+4, z-8)=t(1, -2, 7)

D

(x-1, y+4, z-8)=t(-1, 2, -7)

Vectors, Lines, & Planes

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Cross Product 2 non-parallel (or anti-parallel) vectors lie in one plane. The Cross Product finds a vector perpendicular to that plane.

Vectors, Lines, & Planes

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Cross Product Find a vector perpendicular to plane formed by

Vectors, Lines, & Planes

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84Which of the following is perpendicular to the plane formed by a= (3, -1, 2) and b= (4, -5, 6)

A

B C D

Vectors, Lines, & Planes

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