precalculus chapter 8 page 1home.cvc.org/math/hprecalc/precalcch8.pdf · precalculus chapter 8 page...
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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
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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
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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
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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
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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
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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
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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