VECTORS Wallin

Objectives and Essential Questions Distinguish between basic trigonometric functions (SOH CAH TOA) Distinguish between vector and scalar quantities Add vectors using graphical and analytical methods Essential Questions What is a vector quantity? What is a scalar quantity? Give examples of each.

VECTOR Recall: A VECTOR quantity is any quantity in physics that has BOTH MAGNITUDE and DIRECTION Vector Example Magnitude and Direction Velocity 35 m/s, North Acceleration 10 m/s2, South Force 20 N, East An arrow above the symbol illustrates a vector quantity. It indicates MAGNITUDE and DIRECTION

Vector Properties 3 Basic Vector Properties: Vectors can be moved, order of vectors doesn’t matter Vectors can be added together Vectors can be subtracted from one another

Moving Vectors Vectors can be moved parallel to themselves in a diagram. The angle and magnitude of a vector cannot be changed. This is often called the Tip to Tail Method.

Tip to Tail Method When given two vectors, line up each vector tip to tail. Example: Consider two vectors: one vector is 3 meters east and the other is 4 meters east. Tip 3 meters east 4 meters east Tail

Addition ADDITION: When two (2) vectors point in the SAME direction, simply line them up and add them together. EXAMPLE: A man walks 46.5 m east, then another 20 m east. Calculate his displacement relative to where he started. + 46.5 m, E 20 m, E MAGNITUDE relates to the size of the arrow and DIRECTION relates to the way the arrow is drawn 66.5 m, E

Subtraction SUBTRACTION: When two (2) vectors point in the OPPOSITE directions, simply line them up and subtract them. EXAMPLE: A man walks 46.5 m east, then another 20 m west. Calculate his displacement relative to where he started. 46.5 m, E - 20 m, W 26.5 m, E

Determining Vectors Graphically

Graphing Vectors. Drawing vectors using a ruler and protractor to graphically represent vectors using arrows. Rules: Set a scale, Ex: 1 pace = 1 cm Length of arrow indicates vector magnitude. Use a protractor from the origin to find the angle, we call this angle θ.

Cartesian Coordinate System

Coordinate System North West East South

Determining direction Option to report a direction Report all angles from 0° Report angle from the nearest axis EX: 35°West of North OR 125° What the heck does West of North mean?? The second direction, in this case North, is the direction that you begin facing. The first direction, West in this case, is the direction that you turn. N-face W-turn E S

Graphing Vectors Example Graphically resolve the following vector into its horizontal and vertical components 60 meters at 30º North of East

Graphical Resolution Draw an accurate, to scale vector using a ruler and protractor. Scale: 1 cm = 1 meter N 60.0 m Vector goes a little bit north 30° W E Vector goes a little east We label these components Dx and Dy. S

Resolving Components Use a ruler to measure the length of each component. N Scale: 1 cm = 1 meter 60.0 m Dy = 30.0 cm 30.0 meters 30° W E Dx = 52.0 cm 52.0 meters Your measurement will have slight variations, but should be very close because you drew your vector with care. S

Determining Vectors Mathematically

Math Review Function Abbrev. Description Sine sin opp. / hyp. Cosine cos adj. / hyp. Tangent tan opp. / adj. Use inverse functions to find an angle when triangle side are known. -Use the inverse button on your calculator. Ex: tan -1, cos -1, sin -1

Mathematical Addition of Vectors at Angles Sketch the vectors. Break each vector into X & Y components using trig function Put all X & Y components into a chart with appropriate signs. Add all X & Y components Redraw new triangle**. **if necessary Vector X Y 1 2 3 Total

Mathematical Addition of Vectors at Angles Use Pythagorean Theorem to find the resultant. Use inverse tangent to find the angle with respect to the coordinate system. Write complete answer, including magnitude, unit, angle and direction.

2-D VECTOR Example Example: A man travels 120 km east then 160 km north. Calculate his resultant displacement. FINISH Vector X (km) Y (km) 1 120 2 160 Total the hypotenuse is called the RESULTANT 160 km, N VERTICAL COMPONENT START 120 km, E HORIZONTAL COMPONENT

2-Dimensional VECTORS When two (2) vectors are PERPENDICULAR to each other, you must use the PYTHAGOREAN THEOREM FINISH 160 km, N VERTICAL COMPONENT START 120 km, E HORIZONTAL COMPONENT

NEED A VALUE – ANGLE! Just putting N of E is not good enough (how far north of east ?). We need to find a numeric value for the direction. To find the value of the angle we use a Trig function called TANGENT. 200 km 160 km, N We call the angle theta q 120 km, E So the COMPLETE final answer is : 200 km @ 53.1 °North of East

Example EX: A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. Vector X (m) Y (m) 1 35 2 20 3 -12 4 -6 Total 23 14 12 m, W 6 m, S 20 m, N 35 m, E R 14 m, N q 23 m, E The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST

Example (pg 95) A hiker walks 25.5 km from her base camp at 35° south of east. On the second day, she walks 41.0 km in a direction 65° north of east, at which point she discovers a forest ranger’s tower. Determine the magnitude and direction of her resultant displacement between the base camp and the ranger’s tower.

Consider a problem that asks you to find A – B. Subtracting Vectors Consider a problem that asks you to find A – B. Approach: A – B is the same thing as A + (–B) To adjust the B vector add 180°to the measured angle, thereby “flipping” it Example: 2 meters @ 30° becomes 2 meters @ 210° Then continue vector addition as usual.