Understanding Vectors: Magnitude, Direction, and Operations

1. Vectors

2. Representing Vectors • Vector- a quantity that has both magnitude and direction • Graphical Representation: Direction Magnitude

3. Magnitude to Scale • Scale: 1 cm = 10 m/s The vector measures to be 14 cm long. Therefore the magnitude of the vector is 140 m/s

4. Magnitude to Scale • Conversely, if we are told that a vector has a magnitude of 75 m/s. Drawing the vector to scale, we would draw an arrow 7.5 cm long. 7.5 cm

5. Assigning Magnitude • A vectors magnitude can also be assigned to a vector with an arbitrary length. 20 m/s

6. Direction • The direction of vectors along the x or y axis can be denoted by + or -. • Vectors at an angle can be denoted by an angle measurement relative to the axis: 25° 70° 70° below the x-axis 25° above the x-axis or 250°

7. Direction(continued) • Compass directions can also be used: N Stated as: 30 m/s at 50° North of East 30 m/s or 30 m/s at 40° East of North 50° W E S

8. Moving Vectors • A vector can be moved anywhere as long as the vector magnitude and direction are preserved

9. Vector Operations:Addition • When vectors are added together they produce a new vector that represents the sum of the vectors. We call this new vector the resultant.

10. Adding Parallel Vectors • Adding vectors A and B • When adding vectors arrange them in head-to-tail orientation A = 10 m/s B = 5 m/s A + B Resultant = 15 m/s

11. Commutative Property • Vector Addition is commutative. Two vectors can be added in any order. • A + B = B + A A + B B + A

12. Adding Vectors at an Angle • Add vectors A and B • Again arrange vectors in head-to-tail orientation A = 10 m/s B = 5 m/s A B

13. Adding Vectors at an Angle (continued) • The resultant is then drawn from the tail of the first vector to the head of the last vector. • Solve by Pythagorean Theorem A = 10 m/s B = 5 m/s R

14. Adding Vectors at an Angle (continued) • The angle can then be found using right triangle trigonometry A = 10 m/s  B = 5 m/s R = 11.2 m/s

15. Adding Vectors at Acute and Obtuse Angles • Add vectors A and B • Again arrange vectors in head-to-tail orientation A = 10 m/s B = 5 m/s 30° A B

16. Adding Vectors at Acute and Obtuse Angles (continued) • Draw in the resultant • Solve by Law of Cosines Determine angle by adding together 90 and 30. A = 10 m/s 120° B = 5 m/s C

17. Adding Vectors at Acute and Obtuse Angles (continued) • The angle can then be found using the Law of Sines a = 10 m/s 120° b = 5 m/s c = 13.2 m/s

18. Adding Multiple Vectors • Vector Addition is associative. Three vectors can be added in any order. • (a + b) + c = a + (b + c) • Therefore multiple vectors can be added in any order and you will receive the same resultant.

19. Adding Multiple Vectors (continued) B A • example D C E A B R C D E

20. Subtracting Vectors • Subtract vectors A and B • A – B = A + (-B) A = 10 m/s B = 5 m/s A A = 10 m/s R B -B = 5 m/s R = 5 m/s

21. Components of Vectors • For convenience we would like to classify a vector’s direction as pointing in either the x or y directions. • However, this is not obvious for vectors that do not lie along the x or y-axis. • This is why we have a need to describe how much of the vector is projected in the x and y directions. • This is done through the use of components

22. Components of Vectors (continued) • Given the following vector, determine its components in x and y. • First, align the vector on a coordinate axis. v =20 m/s 35°

23. Components of Vectors (continued) Project lines from the tip of the vector to produce a rectangle.(green) Extend vectors from the tail of the original vector along the axis until they reach the intersection of the projection. (blue andred) y v x

24. Components of Vectors (continued) • Each new vector is a vector in either x or y or an x or y component. The components can then be solved for by using right triangle trigonometry. y v = 20 m/s vy 35° x vx