We use arrows to represent vectors. Vectors have magnitude AND direction. (14m/s west, 32° and falling [brrr!]) Scalars do not have direction, only magnitude. ( 14m/s, 32° ) Vectors can be used to graphically describe an object’s motion. We use arrows to represent vectors. tail tip (head) Vectors are written as: a , b , c, etc.. Or a , b , c, etc..

Vectors can help us find our way – like directions. Image you go to a party. You get directions from your house to the party. N GOOD Directions: Go 3 blocks East Turn and go 3 blocks South Turn and go 7 blocks West. W E Distance = _________ S Directions: Go 3 blocks South Turn and go 4 blocks West. Directions: Go 4 blocks South Turn and go 4 blocks West. Directions: Go 5 blocks South West. BEST N N BETTER W E Distance = _________ W E Displacement = _______ S S

Vectors The longer the arrow, the larger the magnitude. The head of the arrow indicates the direction of travel. Here are examples: N On map coordinates: 1 unit of motion South 5 units of motion East 8 units of motion West 8 5 W E 1 S On x-axis coordinates: 5 units of motion right 8 units of motion left -8 +5

Vector Magnitude The length of the arrow must be an indication of the magnitude of the vector. For example, which makes more sense? A: B: 11m 16m 3m 9m 9m 3m 16m 11m

Vector Magnitude & Scale By using a scale, we can assign values to the length of a drawn vector. For example: if the scale is 1cm = 5m/s then a vector that is drawn 3cm long would represent 15m/s. Knowing this, can you calculate the magnitude of the following vectors? A B C Scale: 1cm = 5 m/s

Vector Direction The direction of the arrow head indicates the direction of travel. We will use many coordinate systems, but for now, let’s stick to a map system (N,S,E,W) N This red arrows represents a vector. It represents motion in the East direction. W E S Which vector represents faster motion? N This green arrows represents a vector. It represents motion in the South direction. W E S

Vector Direction Not all vectors will lie along an axis. We must then specify the angle and direction of the vector. N This vector is 35° North of East 35° W E 27° S This vector is 27° East of South We can graphically find the angle of the vector by using a protractor.

Naming a vector direction can be very tricky . . . Be very careful! This vector is 35° East of North 35° This vector is 35° North of East 35° W E 27° This vector is 27° South of East 27° S This vector is 27° East of South

Graphically Analyzing Vectors Scale: 1cm = 10m/s N We can analyze vectors by looking at their Magnitude and Direction. Magnitude: use a ruler and measure the length of the arrow. Use the scale to calculate the magnitude of the vector. Direction: use a protractor to measure the angle of the arrow relative to a coordinate axis. Give the full direction with respect to that axis. W E 27° S N W E 27° Write out the full vector: 70m/s @ 27° North of East 70m/s @ 63° East of North S OR

Vector Resolution Vectors are not always on the coordinate axis. When this occurs, we need to break down the vector into its x component and its y component. This is called vector resolution. Y component N X component W E Measure to find the magnitude ! S To do this: 1. draw a vertical line at the head of the vector parallel to the vertical axis. 2. draw a horizontal line at the head of the vector parallel to the horizontal axis.

Finding the Resultant Vector Sometimes you may know the X component and Y component of an object’s motion. In this case, you must take those two vectors and substitute them with one vector. This vector is called the Resultant Vector. A Resultant vector is the simplest vector that allows you to represent the same motion. N Y component Measure to find the magnitude ! Measure to find the direction ! W E X component S To do this: 1. draw a vertical line at the head of the X component, parallel to the vertical axis. 2. draw a horizontal line at the head of the Y component, parallel to the horizontal axis. 3. draw the Resultant Vector from the tails of the components to the intersection of the dotted lines.

Vectors can be moved anywhere on your page. All of these vectors have the same magnitude and direction. If you move a vector on your page, it does not change as long as The length and direction of the arrow remains the same. Not the same vector WHY ??

Vector Addition This is the: Tail-to-Tip Method When we add vectors, we just put the tail of one vector onto the head of the other vector. Here’s an example: trip #1: a car travels 30m left. trip #2: a car travels 20m up. trip #3: a car travels 40m right. how can we add the car’s vectors? This is the: Tail-to-Tip Method

Vector Addition Let’s look at some linear vector addition: (in a straight line) If a plane is flying at 100m/s East and encounters a tail wind of 20m/s, how fast is the airplane going? 100 m/s 20 m/s By vector addition: (Tail-to-Tip Method) 120 m/s This is called the Resultant Vector - It goes from the tail of the first vector to the tip of the last vector

Vector Addition Let’s look at some linear vector addition: (in a straight line) If a plane is flying at 100m/s East and encounters a head wind of 20m/s, how fast is the airplane going? 100 m/s 20 m/s By vector addition: (Tail-to-Tip Method) 80 m/s This is called the Resultant Vector - It goes from the tail of the first vector to the tip of the last vector

Vector Addition Now let’s look at some vector addition NOT in a straight line. If a plane is flying at 100m/s East and encounters a crosswind of 20m/s, how fast is the airplane going? 100 m/s 20 m/s You need Pythagorean Theorem To solve this problem ! By vector addition: (Tail-to-Tip Method) Drawn from tail of first vector to head of second vector a2 + b2 = c2 1002 + 202 = c2 10400 = c2 102m/s = c c 20 m/s 100 m/s

This is the “real life” effect of wind vectors on an aircraft: tail wind head wind cross wind Called “wind shear” - this is very dangerous to pilots. This is why it takes about 6 hours to fly from LAX to PHL This is why it takes about 8 hours to fly from PHL to LAX

YES, it has the same Resultant vector ! Vector Addition This is vector addition It is written as: a + b or as a + b (Tail to Tip Method) Resultant vector b a Riddle me this Bartman . . . ???? Is a + b = b + a ???? a a + b b b b + a a YES, it has the same Resultant vector !

Working with more than 2 vectors: By using the Tail-to-tip Method, you could graphically add two or more vectors. C + + = B C A B A We can add as many vectors together as we wish. But, when we want to calculate the Resultant Vector, we must remember that it is the vector going FROM the Tail of the FIRST vector TO the Head of the LAST vector. Head of LAST vector C Resultant Vector B Tail of FIRST vector A SO: + + = R C A B

Examples of Graphical Vector Solutions using Tail-to-Tip Method and Resultant Vectors: A commercial airliner flies 100m East from PHL to Atlantic City Airport. It then flies 1000m South to Charlotte SC. It then flies 2000m West to Las Vegas. It makes its last leg of the trip by flying 2800m North to Green Bay WI. What is the airliner’s displacement? Graphically: The Resultant vector is the displacement. N W E Tip: Use a scale to measure the length of the Resultant Vector to obtain the magnitude. Use a protractor to measure the direction of the Resultant Vector. S