Vectors

Vectors Has both Magnitude and Direction Example: “A bag of candy is located outside the classroom. To find it, displace yourself 20 meters.” What’s the problem?

How about… “A bag of candy is located outside the classroom How about… “A bag of candy is located outside the classroom. To find it displace yourself from the center of the classroom door 20 meters in a direction 30 degrees to the west of north.”

Scale Vector arrow Magnitude and direction Vector Diagrams Scale Vector arrow Magnitude and direction 20 m, 30 degrees West of North

Vectors are expressed as a counterclockwise angle of rotation

Magnitude of a Vector Using the scale (1 cm = 5 miles), a displacement vector that is 15 miles will be represented by a vector arrow that is 3 cm in length. Similarly, a 25-mile displacement vector is represented by a 5-cm long vector arrow. And finally, an 18-mile displacement vector is represented by a 3.6-cm long arrow. See the examples shown below.

Adding Vectors

Example… Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

Practice A Answer R2 = (5)2 + (10)2 R2 = 125 R = 11.2 km

Practice B Answer R2 = (30)2 + (40)2 R2 = 2500 R = 50 km

Using Trig to Determine a Vector’s Direction

Stupid Ways, Important Stuff Sin = o/h Cos = a/h Tan = o/a

Remember the hiker problem from before?

Practice A Tan Ɵ = (5/10) = 0.5 Ɵ = tan-1 (0.5) Ɵ = 26.6 ° Direction of R = 90 ° + 26.6 ° Direction of R = 116.6 °

Practice B Tan Ɵ = (40/30) = 1.333 Ɵ = tan-1 (1.333) Ɵ = 53.1 ° Direction of R = 180 ° + 53.1 ° Direction of R = 233.1 °