Scalars Vectors Examples of Scalar Quantities: Length Area Volume Time A scalar quantity is a quantity that has magnitude only and has no direction in space A vector quantity is a quantity that has both magnitude and a direction in space Examples of Scalar Quantities: Length Area Volume Time Mass Examples of Quantities: Displacement Velocity Acceleration Force

Vector Diagrams Vector diagrams are shown using an arrow Length= magnitude (number) Arrow points toward direction

Tip to Tail Method Adding Vectors To add vectors in this fashion, you must line up the vectors head to tail without changing the direction or the length of each vector: The resultant of the three vectors is a vector that connects the head of the last vector to the tail of the first vector. resultant

Example. Add the following vectors and draw the resultant vector. Steps: Place them head to tail Draw resultant

Example A runner travels 10.0 kilometers due North, 4 kilometers due East, then 2.0 kilometers due South. What is the magnitude (value) of her resultant displacement? a2 + b2 = c2 4 Km 4 Km 2 Km 42 + 82 = c2 10 Km 8 km 80= c2 8.9 = c 8.9 Km

Adding opposing vectors. An observer sees a train moving at a rate of 40 km/hr east and inside the train, a person walks 10 km/hr west. Relative to the observer, how fast is the person walking? 40 km/ hr West 30 km/ hr West 10 km/ hr East The observer will see the person walking at a rate of 30 km/hr to the west.

Identifying Direction A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.) Length = 40 m E W S N 40 m, 50o N of E 60o 50o 40 m, 60o N of W 60o 60o 40 m, 60o W of S 40 m, 60o S of E

Identifying Direction Write the angles shown below by using references to east, south, west, north. E W N 50o S E W S N 45o 500 S of E Click to see the Answers . . . 450 W of N

Remember Trig?!?! B A C q

Easy Way to remember Trig Functions Some Old Hippie, Caught Another Hippie, Tripping On Acid S O H C A H T O A = = =

The height of the building is 57.7 meters Example Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30o. The height h is opposite 300 and the known adjacent side is 90 m. Tan θ = Opp Hyp (90) Tan 30° = Opp 90 (90) (90) Tan 30° = Opp 57.7 = Opp h 300 The height of the building is 57.7 meters 90 m h = (90 m) tan 30o

Vectors have components When any vector is directed at an angle – it can be split into a “x” component and a “y” component. = + NORTHWEST vector NORTHERN vector WESTWARD vector

Finding Components of Vectors A component is the part of a vector that falls along the x or y axis. The x and y components of the vector are illustrated below. x y R q Finding components: x = R cos q y = R sin q

The person is 346 m East and 200 m North of his original location. A person walks 400 m in a direction of 30o N of E. How far is the displacement east and how far north? N The person is 346 m East and 200 m North of his original location. 400 m 30o E The x-component (East) is adj: The y-component (North) is opp: x = R cos q y = R sin q x = 400 cos 30 y = 400 sin 30 x = 346 y = 200