Vectors An Introduction
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 R This is how you draw a vector tail R head
Direction of Vectors Vector direction is the direction of the arrow, given by an angle. This vector has an angle that is between 0 o and 90 o. A x
Vector angle ranges y Quadrant II 90 o < < 180 o Quadrant I 0 < < 90 o Quadrant III 180 o < < 270 o x Quadrant IV 270 o < < 360 o
Direction of Vectors What angle range would this vector have? What would be the exact angle, and how would you determine it? Between 180 o and 270 o B or between - 90 o and -180 o x
Magnitude of Vectors The best way to determine the magnitude (or size) 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? B A 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
opposite The Right Triangle adjacent θ
opposite Pythagorean Theorem hypotenuse 2 = opposite 2 + adjacent 2 c 2 = a 2 + b 2 adjacent θ
opposite Basic Trigonometry functions sin θ = opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent SOHCAHTOA adjacent θ
opposite Inverse functions θ = sin -1 (opposite/hypotenuse) θ = cos -1 (adjacent/hypotenuse) θ = tan -1 (opposite/adjacent) SOHCAHTOA adjacent θ
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 50 o. How wide is the river, and how far is she from the tree in her new location?
Sample problem You are standing erect 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 75 o 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 = A x A A x = A cos A A x x
Vectors: y-component The y-component of a vector is the shadow it casts on the y-axis. sin θ = opposite hypotenuse sin θ = A y A A y = A sin y A y A A y x
Vectors: angle The angle a vector makes with the x- axis can be determined by the components. y It is calculated by the inverse tangent function = tan -1 (A y /A x ) A x A y x
Vectors: magnitude The magnitude of a vector can be determined by the components. It is calculated using the Pythagorean Theorem. R 2 = R x 2 + R y 2 y R R y R x x
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?
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. 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 vectors a) R = 175 meters @ 95 o
Practice Problem Find the x- and y-components of the following vectors b) v = 25 m/s @ -78 o
Practice Problem Find the x- and y-components of the following vectors c) a = 2.23 m/s 2 @ 150 o
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.
Practice Graphical Addition B B A R A + B = R R is called the resultant vector!
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.
The Equilibrant Vector B A -R R A + B = R The vector -R is called the equilibrant. If you add R and -R you get a null (or zero) vector.
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.
Practice Graphical Subtraction -B B C A A - 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 30 o relative to the x axis. Vector C has a magnitude of 50 m and points in a direction of -60 o relative to the x axis. a) Find A + B b) Find A + B + C c) Find A B.
a)
b)
c)
Component Addition of Vectors 1) Resolve each vector into its x- and y- components. A x = Acos B x = Bcos A y = Asin B y = Bsin C x = Ccos C y = Csin etc. 2) Add the x-components (A x, B x, etc.) together to get R x and the y-components (A y, B y, etc.) to get R y.
Component Addition of Vectors 3) Calculate the magnitude of the resultant with the Pythagorean Theorem (R = R x 2 + R y2 ). 4) Determine the angle with the equation = tan -1 R y /R x.
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?
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 V s V w V w V t = V s + V w
Relative Motion V s V w V w V t = V s + V w
Relative Motion V w V s V w V t = V s + V w
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?
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. b) If you paddle west, what is your velocity relative to the shore?
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. 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 0 o.
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?