Download 1.1 Vectors

Math 151: 1.1 Vectors
Two-Dimensional Rectangular and Polar Coordinate Systems
A plane is a two-dimensional space or flat surface that has length and width but no height,
like a floor, a wall, a piece of paper, or the surface of a smart phone. The word
“coordinate” comes from “co-“ meaning “to share” and “ordinate” which comes from Latin
meaning, “to put in order”.
In two dimensions, point P can be located using rectangular or polar coordinates:
Rectangular Coordinates
Polar Coordinates
Superimposing Systems
P
P
P
r
y
x
O
Start at the origin and
move 3 spaces right and 4 spaces
up. Round brackets are used for
rectangular coordinates.
Rotate the initial arm on the circle
of radius 5 through an angle of
approximately 53o . Angled
brackets are used for polar
coordinates.
By superimposing the two
coordinate systems, we can relate
x, y, the radius and the angle.
Complex Plane
We can also plot P as the complex
number z = 3 + 4i , in the Complex
Plane using rectangular coordinates.
P or
•
The horizontal axis is a numberline
for the real part of z: Re z = 3 .
•
The vertical axis is a numberline
for the imaginary part of z:
Im z = 4 .
()
()
This representation of
a complex number
z = x + iy or
z = cosθ + i sinθ is
given on the unit
circle for r = 1 .
Euler’s formula
expresses
cosθ + i sinθ = e iθ .
Two-Dimensional Vectors
Quantities for which only “how much” matters are called scalars and those for which
direction also matters are called vectors. In two dimensions, a vector is a directed line
segment, like an arrow from one point to another in the Cartesian plane.
© Raelene Gibson 2017
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Math 151: 1.1 Vectors
A vector:
• describes the horizontal and vertical changes from an initial point to a terminal point:
these are the rectangular components, Δx and Δy .
• describes “how much” (length or magnitude, r) and “in which direction” ( θ measured in
standard position).
• Standard position of a vector is translated so that the initial point is ___________.
• Two vectors are equivalent if they have the same _______________ components
(____ and ____) or the same _______________ components (____ and ____).
r
Rectangular Components, v = ⎡⎣ Δx,Δy ⎤⎦
•
Δx = xf − xi (the run)
•
Δy = y f − y i (the rise)
from initial point P to terminal point Q
P
r
Polar Components, v = r,θ
•
•
r (magnitude)
θ (direction)
Q
Rectangular to Polar:
• r 2 = Δx 2 + Δy 2
•
tanθ ref =
Polar to Rectangular:
• Δx = r cosθ
• Δy = r sinθ
Δy
Δx
Exercises: Sketch and fully label each vector.
1. Convert to polar components:
(a) ⎡ −4,4 3 ⎤
⎣
⎦
(b) ⎡⎣0,−9.8 ⎤⎦
(c) z = −12 − 5i
2. Convert to rectangular components: (a) 10, π6 rad
(b) 1,300o
(c) mg,270o
r
Exercise 3: Let v be the vector from point A 3,1 to point B −1,4 .
r
r
a) Sketch A, B and v , make a right triangle and determine/label the components of v .
r
b) Sketch v using P −2,−3 as a terminal point. Determine the new initial point.
r
c) Sketch v using a,b as an initial point. Determine the new terminal point.
r
d) Sketch v in standard position, using the origin as the initial point.
( )
(
( )
© Raelene Gibson 2017
(
)
)
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Math 151: 1.1 Vectors
Exercise 4: Are the two vectors equivalent:
r
r
v = 10, 76π and u , from A 8 3,4 to B 3 3,−1 ? Explain.
(
)
(
)
Vector Operations
The two basic vector operations are (1) Scalar Multiplication and (2) Vector Addition. One
type of vector multiplication called the Dot (or Scalar) Product and a second type of vector
multiplication called the Cross (or Vector) Product, which is studied in Calculus 3 and
Physics.
Scalar Multiplication
• multiplying a vector by a constant scalar
r
• horizontal and vertical scaling by a constant: k ⋅ v
• resizing the vector
• making similar triangles
• changes the magnitude, Δx and Δy
• keeps angles (direction) the same
• if k < 0 , then the vector has been reflected (both
horizontally and vertically) which is the same as a
180o rotation about the initial point.
• the negative of a vector reverses the direction/swaps
the initial and terminal points.
Vector Addition
• the vector which is the sum of two other vectors is
r r
called the resultant vector, u + v
• add vectors by rectangular (not polar) components
r
r
• if u is the vector from point A to point B and v is the
r r
vector from point B to point C, then u + v is the vector
from point ___ to point ___ .
Linear Combination of Vectors and Standard Unit Vector (or Basis Vector) Notation
• a unit vector is a vector that has a magnitude or length of 1 (like the radius of the unit
circle is 1).
• the “standard or basis vectors” in two-dimensions point in the directions of 0o (along
the positive x - axis) and 90o (along the positive y – axis) and are called iˆ and jˆ . The
“hat” notation indicates that the vector is a unit vector.
© Raelene Gibson 2017
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Math 151: 1.1 Vectors
Complete the table:
Symbol
Polar Form
iˆ
r,θ = 1,0o
jˆ
•
•
•
Rectangular Form
Graph in xy - plane
⎡⎣ Δx,Δy ⎤⎦ = ⎡⎣0,1⎤⎦
r
any vector can be expressed as v = aiˆ + bjˆ , a sum of a scaling of iˆ and a scaling of jˆ
⎡⎣ −3,4 ⎤⎦ = −3iˆ + 4 jˆ = 5,π − tan−1 34
the benefit of this notation is that vectors in this notation can be treated using regular
rules of algebra
r
Exercise 5: Express the vector w from A 2,3 to B −1,−1 .
( )
a)
b)
c)
d)
e)
(
)
graphically (draw three equivalent vectors on the same grid).
in rectangular components (why don’t we use the word “coordinates” here?).
in standard vector notation (and indicate the standard vectors on the graph).
in polar components.
r
r
Determine a vector v with the magnitude of −4iˆ + 4 jˆ in the direction of w .
Exercise 6: Can vector addition and scalar multiplication of vectors be performed on
vectors given in:
a) rectangular form? Explain or give a counter-example.
b) polar form? Explain or give a counter-example.
r
r
Exercise 7: The vectors a = 2,−1 and b = −3,2 are given in rectangular form.
r
a) What is the length of a ?
r
b) Express a in standard vector notation.
r
c) Write a vector in standard vector notation that is in the same direction as a but has a
length of 1. rWhat is
a good symbol for this vector?
r
r
r r
d) Sketch a + b , a − b and 5a .
r
r
e) Express a and b in standard vector notation.
r
r
r
f) Determine the vector c = 5a − 2b in standard vector and in polar forms.
© Raelene Gibson 2017
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Math 151: 1.1 Vectors
Checkpoint:
Match the transformation with its definition and circle the transformation(s) that DO NOT
change vectors.
Resizing/changing magnitude but not direction.
Reflecting
Changing the signs/direction of either Δx or Δy or both.
Rotating
Moving all points the same distance in the same direction.
Scaling
Changing the direction/angle about a point but not the magnitude.
Translating
r
Exercise 8: Let v = ⎡⎣ −2,6 ⎤⎦ = −2iˆ + 6 jˆ .
r
r
a) Sketch v and state in polar form, v ,θv = v,θv = magnitude, direction .
!
b) Write an equation representing the family of lines that are parallel to v .
c) How is the slope of the line related to the angle of inclination of the line?
!
d) Determine the distance between any pair of initial and terminal points on v .
e) Determine all unit vectors that are
r
(i)
parallel to v
r
(ii)
anti-parallel to v
r
(iii)
orthogonal to v
r
f) Determine all vectors that are parallel to v in standard vector notation.
g) Write the equation of the family of lines that are parallel to the vector in (b).
© Raelene Gibson 2017
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Math 151: 1.1 Vectors
Vectors in Physics
r
r
Position r or s is a vector that
• describes location relative to the origin (or another defined reference point)
• has horizontal and vertical components, measured in meters
• has magnitude in meters (known as magnitude of position) and direction
Displacement is a vector that
r
• describes change in position, Δr
• has horizontal and vertical components, measured in meters
• has magnitude in meters (known as magnitude of displacement) and direction
Distance versus Displacement
• Distance is a scalar and displacement is a vector.
• In general, the scalar quantity distance IS NOT the magnitude of displacement.
• In the special case that a moving particle does not change direction, the distance is
equal to the magnitude of displacement.
r
Velocity v is a vector that
• describes the (average or instantaneous) rate of change of position w.r.t. time
• in 1D, velocity is the slope (secant or tangent) of the position-time graph
• has horizontal and vertical components, measured in m/s
• has magnitude in m/s (known as “speed”) and direction
r
Acceleration a is a vector quantity that
• describes the (average or instantaneous) rate of change of velocity w.r.t. time.
• has horizontal and vertical components, measured in m/s2
• has magnitude in m/s2 (known as “magnitude of acceleration”) and direction
Velocity versus Speed
• Speed is a scalar: “how fast” a particle is travelling.
• 1D motion means that a particle moves along a line (like a wire); 2D motion means
that a particle moves in a plane (like the xy – plane).
• Speed is the slope of the 1D distance-time graph.
• Velocity is a vector: “how fast” AND “where” the particle is traveling.
• Velocity is the slope of the 1D position-time graph.
• Speed is the magnitude of velocity.
• It is unusual for the magnitude of a vector to be given a special name. For example,
the magnitude of position is “magnitude of position”; the magnitude of force is
“magnitude of force”; the magnitude of acceleration is “magnitude of acceleration”;
but the magnitude of velocity is called “magnitude of velocity” or “speed”.
• Because the magnitude of velocity has been given a special name, speed seems to
have important status that misleads us to as to its meaning. Speed means nothing
more than the “magnitude of velocity”. In fact, speed is the only special name given
to any vector’s magnitude (that I can think of right now); I think we might be less
confused by “speed” if we started referring to it as ”magnitude of velocity”.
Gravity is
• the attraction or pull between any two bodies that have mass, not always involving
the earth, as summarised by Newton’s Universal Law of Gravitation.
r
• For clarity, we will refer to the force of gravity (on Earth) as weight Fg (in newtons)
r
r
and the acceleration due to gravity as ag , where ag = g = 9.8 m/s2 ≈ 10 m/s2 is the
r
magnitude and the direction of a is 270o (straight down).
© Raelene Gibson 2017
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Math 151: 1.1 Vectors
r
Exercise 9: Express ag , the acceleration due to gravity on Earth, in
a) polar form (use angled brackets)
b) rectangular form (use rectangular brackets).
c) standard vector form.
Exercise 10: Determine the direction of each vector.
r
r
r
a) a = −π iˆ + π jˆ
b) b = −9.8 jˆ
c) c = 4iˆ − 4 3 jˆ
r
d) d = −100iˆ
Exercise 11: Determine the exact magnitude and direction of the resultant force vector of
r
r
r
F1 = 8 N, 120o and F2 = 6 N, 240o . Call the resultant Fnet . Include a sketch.
Exercise 12: On a windless day, Ali runs across an open field at 6 km/hr at a direction of
150o . Today, the wind blows with a constant velocity of 12 km/hr at a direction of 270o .
a) Determine Ali’s velocity when running with today’s wind.
b) How many degrees off his original course does the wind push Ali?
c) Determine Ali’s net speed. Is this a scalar or a vector quantity?
© Raelene Gibson 2017
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Math 151: 1.1 Vectors
Answer on looseleaf paper. Organisation, completeness and neatness count.
Exercise 13:
Two ropes are used to suspend an 80 lb object from the ceiling. One rope forms an angle
of depression of 60o and the other forms an angle of depression of 30o , both from the
ceiling. Determine the tensions T1 and T2 in each rope. Express your answer in both unit
vector notation and polar form. Include a sketch in your solution.
Exercise 14:
Two ropes are used to suspend an 10 kg object from the ceiling. One rope forms an angle
of depression of 60o and the other forms an angle of depression of 30o , both from the
ceiling. (a) Determine the tension in each rope. Use the same instructions from #13.
(b) Repeat the problem changing 10 kg to m and 60o , 30o to acute angles a and b .
Exercise 15:
An object is launched on Earth at an initial velocity of 50 m/s at an angle of 30o from a
height of 30 meters. This 2D motion is a generalisation of 1D free-fall (when the only force
acting on the object is gravity) is known as projectile motion. The position, velocity and
height equations for projectile motion are the same as for 1D free-fall, except that they are
valid for both horizontal and vertical components.
Position
Velocity
Acceleration
1D free-fall
h t = 21 ag t 2 + v ot + ho
()
()
v t = ag t + v o
()
a t = ag
r
r t
r
(1) r t
r
v t
r
(2) v t
r
a t
r
(3) a t
2D Projectile Motion
r 2 r
r
= ag t + v ot + ro
()
( ) = ( a t + v t + r ) iˆ + ( a t
r
r
( ) = a t +v
( ) = (a t + v ) iˆ + (a t + v ) jˆ
r
( )=a
( ) = a iˆ + a jˆ
1
2
2
1
2
gx
g
ox
ox
1
2
gy
oy
gy
2
)
+ v oy t + roy jˆ
o
gx
ox
g
gx
gy
a) Given the launch information above, write the following vectors in both rectangular
r r
r
polar forms: ag , v o and ro . Use ag = g = 10 m/s2 .
b) Write Cartesian equations and sketch graphs for each of the 6 functions of time: (1)
rx = x vs. t and (2) ry = y vs. t; (3) v x vs. t and (4) v y vs. t; (5) ax vs. t and (6) ay vs. t.
c)
d)
e)
f)
g)
h)
i)
j)
k)
Neatly label 3 points on each graph for the times that correspond to launching,
reaching maximum height and landing.
What is the time domain (for the object to be in projectile motion/free-fall)?
What is the maximum height reached by the object, and when did the object reach it?
What is the horizontal position when the object is at its maximum height?
How far away horizontally from launching did the object land?
What is the speed of the object at t = 5 seconds?
What is the total vertical distance traveled by the object?
What is the displacement of the object from beginning to end of the launch?
What is the horizontal position of the object when the vertical component of the velocity
is -25 m/s?
Write an equation for the vertical position of the object as a function of the horizontal
position.
© Raelene Gibson 2017
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