Vectors 7.4

1. Vectors7.4 JMerrill, 2007 Revised 2009

2. Definitions • Vectors are quantities that are described by direction and magnitude (size). • Example: A force is a vector because in order to describe a force, you must specify the direction in which it acts and its strength. • Example: The velocity of an airplane is a vector because velocity must be described by direction and speed. • The vector is the zero vector. It has no direction.

3. Representation of Vectors • The velocities of 3 airplanes, two of which are heading northeast at 700 knots, are represented by u and v u v We say u = v to indicate both planes have the same velocity. w ≠ u or v, why? The direction is different. w

4. Magnitude • The magnitude of vector v is represented by the absolute value of v. • In the previous example, |u| = 700, |v|= 700, and |w| = 700. We know that |u|=|v|=|w|, but u ≠ w, and v ≠ w, why? • The direction is not the same!

5. Addition of Vectors A 10 B 5 C

6. Addition of Vectors • Addition is commutative, so a + b = b + a • The vector sum is called the resultant. b b+a a a a+b b

7. Vector Subtraction • A negative vector has the same magnitude, but in the opposite direction. • v + (-v) = 0 • v – w means v + (-w) v -v v w

8. Multiples of Vectors -v v -2v 2v 3v -3v

9. You Do • Let u = and v = • Find 3u + 2v • Find ½ u + 4v • Find u – 2v

10. Scalar Multiplication • Real numbers are often referred to as scalars. • When we multiply a vector by a scalar, we use the same rules that we are familiar with: k(v + w) = kv + kw k(mv) = kmv

11. Component Form • From the tail to the tip of vector v, we see: • A 2 unit change in the x-direction, and • A -3 unit change in the y-direction. • 2 and -3 are the components of v. • When we write v = we are expressing v in component form. 2 3

12. You can count the number of spaces to get the component form or, you can subtract the coordinates. IT IS ALWAYS B – A! The magnitude of vector AB is found using the distance formula: Component Form (x2,y2) (y2 – y1) (x2 – x1) (x1,y1)

13. Example • Given A(4, 2) and B(9, -1), express in component form. Find

14. Vector Operations with Coordinates • Vector Addition • v + u = • Vector Subtraction • v - u = • Scalar Multiplication • kv =

15. Example • If u = and v = , find: • u + v • u – v • 2u – 3v

16. Drawing • Draw a parallelogram if you have a force. • Draw using tip-to-tail if you have a change of course. • ALWAYS, ALWAYS, ALWAYS make your drawing in proportion. • And remember, heading/bearing/compass direction is always measured clockwise from magnetic north!

17. Example • A force of 20N (20 Newtons) is pulling an object east and another force of 10N is pulling the object in the compass direction of 150o. Find the magnitude and direction of the resultant force. • Let O = object (and make it at the origin for ease of computation)

18. 10N Draw what you know Draw the parallelogram Draw the resultant—that’s what you’re looking for! Use Law of Sines/Cosines to find magnitude and direction. We will use only one of the triangles. Example 150o 20N x 60o x

19. 10N We know the obtuse angle = 120 degrees. Let r = resultant. Use the Law of Cosines to find r. Now find angle x: Example 120o x 20N 90o + 19.1o = 109.1o So the resultant force is 26.46N in a direction of 109.1o

20. You Do: • An airplane has a velocity of 400mph southwest. A 50mph wind is blowing from the west. Find the resultant speed and direction of the plane. • We’re changing direction, so use tip-to-tail. • The plane’s resultant velocity is about 366mph on a course of approximately 219.5 degrees