vectors an introduction with components september 19, 2013
TRANSCRIPT
Vectors
An Introduction with Components
September 19, 2013
There are two kinds of quantities…
SCALARS are quantities that have magnitude only, such as position speed time mass
VECTORS are quantities that have both magnitude and direction, such as displacement velocity acceleration
Notating vectors
This is how you notate a vector…
R headtail
This is how you draw a vector…
RBooks usually write vector names in bold.
RYou would write the vector name with an arrow on top.
Direction of Vectors
Vector direction is the direction of the arrow, given by an angle.
Draw a vector that has an angle that is between 0o and 90o.
Ax
Vector angle ranges
x
y
Quadrant I0 < < 90o
Quadrant II90o < < 180o
Quadrant III180o < < 270o
Quadrant IV
270o < < 360o
Direction of Vectors
What angle range would this vector have? What would be the exact angle, and how
would you determine it?
Bx
Between 180° and 270°
or between-270° and -180°
Magnitude of Vectors
The best way to determine the magnitude of a vector is to measure its length.
The length of the vector is proportional to the magnitude (or size) of the quantity it represents.
Sample Problem
If vector A represents a displacement of three miles to the north, then what does vector B represent? Vector C?
A
B
C
Sample Problem
If vector A represents a displacement of three miles to the north, then what does vector B represent? Vector C?
A
B
C
Equal Vectors
Equal vectors have the same length and direction, and represent the same quantity (such as force or velocity).
Draw several equal vectors.
Inverse Vectors
Inverse vectors have the same length, but opposite direction.
Draw a set of inverse vectors.
A
-A
The Right Triangle
θ
oppo
site
adjacent
hypotenuse
Pythagorean Theorem
hypotenuse2 = opposite2 + adjacent2
θ
oppo
site
adjacent
hypotenuse
Basic Trigonometry functions
sin θ = opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent
θ
oppo
site
adjacent
hypotenuseSOHCAHTOA
Inverse functions
θ = sin-1(opposite/hypotenuse) θ = cos-1(adjacent/hypotenuse) θ = tan-1(opposite/adjacent)
θ
oppo
site
adjacent
hypotenuseSOHCAHTOA
Sample problem A surveyor stands on a riverbank directly across the river from a tree on
the opposite bank. She then walks 100 m downstream, and determines that the angle from her new position to the tree on the opposite bank is 50o. How wide is the river, and how far is she from the tree in her new location?
Sample problem• A surveyor stands on a riverbank directly across the river from a
tree on the opposite bank. She then walks 100 m downstream, and determines that the angle from her new position to the tree on the opposite bank is 50o. How wide is the river, and how far is she from the tree in her new location?
Sample problem• You are standing at the very top of a tower and notice that
in order to see a manhole cover on the ground 50 meters from the base of the tower, you must look down at an angle 75o below the horizontal. If you are 1.80 m tall, how high is the tower?
Sample problem• You are standing at the very top of a tower and notice that
in order to see a manhole cover on the ground 50 meters from the base of the tower, you must look down at an angle 75o below the horizontal. If you are 1.80 m tall, how high is the tower?
Vectors: x-component
The x-component of a vector is the “shadow” it casts on the x-axis.
cos θ = adjacent ∕ hypotenuse cos θ = Ax ∕ A
Ax = A cos
A
x
Ax
Vectors: y-component
The y-component of a vector is the “shadow” it casts on the y-axis.
sin θ = opposite ∕ hypotenuse sin θ = Ay ∕ A
Ay = A sin
A
x
y
Ay Ay
Vectors: angle
The angle a vector makes with the x-axis can be determined by the components.
It is calculated by the inverse tangent function
= tan-1 (Ay/Ax)
x
y
Rx
Ry
Vectors: magnitude
The magnitude of a vector can be determined by the components.
It is calculated using the Pythagorean Theorem.
R2 = Rx2 + Ry
2
x
y
Rx
Ry
R
Conceptual Practice Problem
Can the magnitude of a component of a vector be greater than the vector’s magnitude?
Explain…
Practice Problem You are driving up a long inclined road. After 1.5 miles you
notice that signs along the roadside indicate that your elevation has increased by 520 feet.
a) What is the angle of the road above the horizontal?
b) How far do you have to drive to gain an additional 150 feet of elevation?
Practice Problem Find the x- and y-components of the following
vectorsa) R = 175 meters @ 95o
b) v = 25 m/s @ -78o
c) a = 2.23 m/s2 @ 150o
Practice ProblemsA water molecule is shown in the diagram.
The distance from the center of the oxygen atom to the center of the hydrogen atom is 0.96 angstroms, and the angle between the hydrogen atoms is 104.5°.
Find the center-to-center distance between the hydrogen atoms.
104.5°
0.96
Ǻ
Vectors
Graphical addition and subtraction
Graphical Addition of Vectors
1) Add vectors A and B graphically by drawing them together in a head to tail arrangement.
2) Draw vector A first, and then draw vector B such that its tail is on the head of vector A.
3) Then draw the sum, or resultant vector, by drawing a vector from the tail of A to the head of B.
4) Measure the magnitude and direction of the resultant vector.
A
B
RA + B = R
Practice Graphical Addition
R is called the resultant vector!
B
The Resultant and the Equilibrant
The sum of two or more vectors is called the resultant vector.
The resultant vector can replace the vectors from which it is derived.
The resultant is completely canceled out by adding it to its inverse, which is called the equilibrant.
A
B
R A + B = R
The Equilibrant Vector
The vector -R is called the equilibrant.If you add R and -R you get a null (or zero) vector.
-R
Graphical Subtraction of Vectors
1) Subtract vectors A and B graphically by adding vector A with the inverse of vector B (-B).
2) First draw vector A, then draw -B such that its tail is on the head of vector A.
3) The difference is the vector drawn from the tail of vector A to the head of -B.
A
B
A - B = C
Practice Graphical Subtraction
-B
C
Practice Problem
Vector A points in the +x direction and has a magnitude of 75 m. Vector B has a magnitude of 30 m and has a direction of 30o relative to the x axis. Vector C has a magnitude of 50 m and points in a direction of -60o relative to the x axis.
a) Find A + Bb) Find A + B + Cc) Find A – B.
a)
b)
c)
Vectors
Component addition and subtraction
Component Addition of Vectors
1) Resolve each vector into its x- and y-components.
Ax = Acos Ay = Asin
Bx = Bcos By = Bsin
Cx = Ccos Cy = Csin etc.
2) Add the x-components (Ax, Bx, etc.) together to get Rx and the y-components (Ay, By, etc.) to get Ry.
Component Addition of Vectors
3) Calculate the magnitude of the resultant with the Pythagorean Theorem (R = Rx
2 + Ry2).
4) Determine the angle with the equation = tan-1 Ry/Rx.
Practice Problem
In a daily prowl through the neighborhood, a cat makes a displacement of 120 m due north, followed by a displacement of 72 m due west. Find the magnitude and displacement required if the cat is to return home.
Practice Problem
If the cat in the previous problem takes 45 minutes to complete the first displacement and 17 minutes to complete the second displacement, what is the magnitude and direction of its average velocity during this 62-minute period of time?
Vectors
Relative MotionSeptember 2013
Relative Motion
Relative motion problems are difficult to do unless one applies vector addition concepts.
Define a vector for a swimmer’s velocity relative to the water, and another vector for the velocity of the water relative to the ground. Adding those two vectors will give you the velocity of the swimmer relative to the ground.
Relative Motion
Vs
Vw
Vt = Vs + Vw
Vw
Relative Motion
Vs
Vw
Vt = Vs + Vw
Vw
Relative Motion
Vs
Vw
Vt = Vs + Vw Vw
Practice Problem
You are paddling a canoe in a river that is flowing at 4.0 mph east. You are capable of paddling at 5.0 mph.
a) If you paddle east, what is your velocity relative to the shore?
b) If you paddle west, what is your velocity relative to the shore?
c) You want to paddle straight across the river, from the south to the north.At what angle to you aim your boat relative to the shore? Assume east is 0o.
Practice Problem
You are flying a plane with an airspeed of 400 mph. If you are flying in a region with a 80 mph west wind, what must your heading be to fly due north?