Precalculus Chapter 8 Page 1

Section 8.1 – Geometric Vectors Objectives: 1. To add and subtract vectors geometrically.

I. Definitions

A. Vector is a _____________, or ____________ distance, that has both _____________ and _________________. A ___________ line ____________ represents a __________.

B. _______________

v!( ) -- the __________ of the directed line segment.

C. _____________ Position – ____________ point (tail) is at the ____________. D. Amplitude – the _____________ angle between ________________ x-axis and the _________ when in standard position. E. ______________ – the _________ of two or more vectors

II. Vector Addition A. Method:

1. 2. 3.

B. Examples 1. Find the sum of u

! and v!

. 2. Jan was rowing directly across a river at the rate of 5-mph the current was flowing at a rate of 3.5-mph. Draw a vector representing the path of the boat.

III. More Definitions A. _____________ vectors – same _____________ but _______________ in direction. B. _____________ – multiplies the vector’s ______________. (scalars only have magnitude, no direction) C. _______________ vectors – vectors that have the __________ or ___________ directions. D. Examples:

1. u!+3v!

2. u!− v!

IV. Components A. Definition – _____________ that were ________ together to form a ______________ vector. B. Most useful – ______________ and _______________ components. C. Example – Draw the horizontal and vertical components for u

!+ v!

Homework: p.490 – 2, 14-17 all, (18-30)/3, 31-34 all, 49-52 all

tail

head

Precalculus Chapter 8 Page 2

Section 8.2 – Algebraic Vectors Objectives: 1. To find ordered pairs that represent vectors 2. To add, subtract, multiply, and find the resulting vector.

I. Representation of a vector as an ordered pair.

A. An ordered pair that represents the vector P1P2

! "!!! from standard position is _________________.

B. The magnitude -- P1P2

! "!!!=

C. Example: 1. Find the ordered pair that represents the vector from the points (5,4)A and (0, 3)B − .

2. Find AB! "!!

II. Vector Operations A. Addition -- a

!+b!=

B. Subtraction -- a!−b!=

C. Scalar Multiplication -- ka!=

D. Examples: If u!= 1,−4 and

v!= 0,8 find:

1. u!+ v!

2. u!−3v!

WARNING – Know when you are dealing with vectors or points!!

III. Components and Unit Vectors

A. Horizontal = B. Vertical = C. Unit Vectors

1. i!

represents a unit in the _____________ direction. 2. j!

represents a unit in the _____________ direction.

3. Vector a!

can be represented as a1i!+ a2 j!= a!

D. Examples

1. Vector a!

has a magnitude of 4.3-cm and amplitude of 51°. Find the vertical and horizontal components. 2. Two forces are acting on an object. One is 6-N at an angle of 70° and the other is 10-N at an angle of 115°. What is the resulting vector? 3. Write AB

! "!! as the sum of unit vectors for the points (5,4)A and (0, 3)B − .

Homework: p.497 – (15-42)/3, 44, 47, 49, 53, 55, 57

Precalculus Chapter 8 Page 3

Section 8.3 – Vectors in 3-D Space Objectives: 1. To graph points and vectors in 3-D Space 2. To add, subtract, and find the resulting vector.

I. 3-D Graphing

A. 3-D Coordinate System B. Example:

1. Locate (2,-3,4) 2. Locate (-5,3,-2) 3. Draw 2, 3,4−

4. Draw 5,3, 2− −

II. Representation of a vector A. An ordered pair that represents the vector P1P2

! "!!! from standard position is

2 1 2 1 2 1, ,x x y y z z− − − . B. The magnitude --

P1P2

! "!!!=

C. Example:

1. Find the ordered pair that represents the vector from the points ( 2, 5,0)A − − and (3,1,8)B .

2. Find

AB! "!!

Homework: p.502 – 1, 12-14 all, (15-33)/3, 37, 43, 44, 49, 50

Precalculus Chapter 8 Page 4

Section 8.4 – Perpendicular Vectors Objectives: 1. To find the inner (dot) and cross products of two vectors. 2. To determine if two vectors are perpendicular. 3. To find a 3-D vector that is perpendicular to another 3-d vector.

I. Dot Product

A. Definition – If !a a1,a2 and

!b b1,b2 are two vectors, the dot product of

!a and !b is defined

!a i!b =

B. Theorem – If !a i!b = 0 , then

!a and !b are ________________ vectors.

C. Example:

1. Find the dot product of !a and

!b and

!a and !c if

!a = 2,−5 ,

!b = 4,1 , and

!c = 10,4 .

Is either pair of vectors perpendicular? Show by graphing that this is true. 2. Find the inner product of

!v = 2,−3,−4 and

!w = 8,3,2

II. Cross Product A. Definition – If

!a a1,a2 ,a3 and

!b b1,b2 ,b3 , then the cross product of is defined

!a×!b =

B. Theorem – The cross product of two 3-D vectors is a 3-D vector that is ________________ to each of the given 3-D vectors. C. Example:

Find the cross product of !a and

!b , if

!a = −4,1,0 and

!b = 5,4,−2 . Verify that the

resulting vector is perpendicular to !a and

!b .

Homework: p.509 – 11-25 odds, 29, 33, 36, 41, 42, 44-47 all

Precalculus Chapter 8 Page 5

Section 8.5 – Applications with Vectors Objectives: 1. To solve problems using vectors and right triangle trigonometry.

Examples: 1. A 4-kg mass is sliding down a plane inclined at an angle of 20° with the horizontal.

a) Draw and label a diagram that represents the forces at work. b) Find the force that propels the object down the slope, given that wF mg= , where 29.8g m s= c) Use the formula F ma= to find the objects acceleration.

2. A farmer and his son are removing a large bolder from a well. The are pulling on ropes attached to the bolder. The ropes make a 40° angle. If the farmer is pulling with 115-N and his son with 105-N, what is the magnitude of the net force? 3. Two wires of equal length attached at its corners and tied to a single hook in the wall support a painting. The wires make a 70° angle with one another. What is the tension on each corner if the painting weighs 7.5-lbs.?

Homework: p.518 – 1-9 all, 11-31 odds, 33-34, 36-38 all

35° t

Rock

Precalculus Chapter 8 Page 6

Section 8.6 – Parametric Equations Objectives: 1. To write parametric equations. 2. To graph parametric equations.

A. Definition – A function that relates two variables to a third independent variable called parameter. B. Examples

1. Do Exploration p.526

2. Graph: 4 33 5

x ty t= +

= −

3. Write the parametric equation for the line 1 72

y x= + .

Hints: a) What is the independent variable in this equation? b) What is the independent variable in parametric equations?

4. Write the parametric equation in #2 in linear form.

Homework: p.524 – 18-30 all, 31b,c, 32-35 all, 40-46 all

t x y -1 0 1

−9 −6 −3 3 6 9

−6

−4

−2

2

4

6

Precalculus Chapter 8 Page 7

Section 8.7 – Using Parametric Equations to Model Motion Objectives: 1. To model the motion of a projectile using parametric equations. 2. To solve problems related to the motion of a projectile, its trajectory, and range.

I. Projectiles

A. Definition – ________ that are _________________. B. _________________ – the path of the projectile C. Two components

1. _____________ a) ______________ will act on the vertical velocity b) Velocity is ________ and ____________ at the beginning c) Velocity is ___________ at the _________ of the trajectory d) Velocity is ___________ in the ______________ direction as it falls e) When the ball returns to the ground, the vertical velocity is the vertical velocity as when it was launched, but opposite.

2. _____________ a) Horizontal distances that projectiles travel is called ______________. b) Horizontal velocity is ___________________.

II. Parametric Equations for the path of a projectile A. Vertical Component: B. Horizontal component: C. Examples

1. A football player kicks a ball with an initial velocity of 40 ft/s at an angle of 40° with the horizontal. After 0.8 seconds, how far has the ball traveled horizontally and vertically? 2. An archer shoots an arrow with an initial velocity of 60 m/s at an angle of 4.8° with the horizontal at a target 50 meters away. If the bow is held 1.4 meters off the ground, how far off the ground will the arrow be when it hits the target?

Homework: p.531 – 1-17 all, 20, 21, 23-29 all