VECTORS
Objectives and Essential Questions Objectives 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 Velocity Acceleration Force Magnitude and Direction 35 m/s, North 10 m/s 2, South 20 N, East x, v, a, F An arrow above the symbol illustrates a vector quantity. It indicates MAGNITUDE and DIRECTION
Vector Properties 3 Basic Vector Properties: 1. Vectors can be moved, order of vectors doesn t matter 2. Vectors can be added together 3. 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 66.5 m, E MAGNITUDE relates to the size of the arrow and DIRECTION relates to the way the arrow is drawn
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: 1. Set a scale, Ex: 1 pace = 1 cm 2. Length of arrow indicates vector magnitude. 3. Use a protractor from the origin to find the angle, we call this angle θ.
Cartesian Coordinate System
Coordinate System North West East South
S Determining direction Option to report a direction 1. Report all angles from 0 2. 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
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. N Scale: 1 cm = 1 meter Vector goes a little bit north W 30 Vector goes a little east E S We label these components D x and D y.
Resolving Components Use a ruler to measure the length of each component. N Scale: 1 cm = 1 meter Dy = 30.0 cm 30.0 meters W S 30 Dx = 52.0 cm 52.0 meters E Your measurement will have slight variations, but should be very close because you drew your vector with care.
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 1. Sketch the vectors. 2. Break each vector into X & Y components using trig function 3. Put all X & Y components into a chart with appropriate signs. 4. Add all X & Y components 5. Redraw new triangle**. **if necessary Vector X Y 1 2 3 Total
Mathematical Addition of Vectors at Angles 6. Use Pythagorean Theorem to find the resultant. 7. Use inverse tangent to find the angle with respect to the coordinate system. 8. 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. Vector X (km) Y (km) 1 120 0 2 0 160 Total 120 160 the hypotenuse is called the RESULTANT FINISH 160 km, N VERTICAL COMPONENT S T A R T 120 km, E HORIZONTAL COMPONENT
2-Dimensional VECTORS When two (2) vectors are PERPENDICULAR to each other, you must use the PYTHAGOREAN THEOREM c a b c a b 2 2 2 2 2 c resultant 120 160 c 200km 2 2 FINISH 160 km, N VERTICAL COMPONENT S T A R T 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 We call the angle theta 120 km, E 160 km, N Tan opposite side adjacent side 160 120 1.333 Tan 1 (1.333) 53.1 o 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. 6 m, S 12 m, W 20 m, N Vector X (m) Y (m) 1 35 0 2 0 20 3-12 0 4 0-6 Total 23 14 35 m, E R 23 m, E 14 m, N R 14 2 23 14 Tan.6087 23 1 Tan (0.6087) 2 26.93m 31.3 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.
Approach: Subtracting Vectors Consider a problem that asks you to find A B. 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.