1. Vectors AdditionGraphical && Subtraction Analytical

2. What is a Vector? A quantity that requires magnitude and direction to completely describe it is a vector. An arrow is used to represent a vector. The length of the arrow represents the magnitude. Some examples of vectors are: displacement, velocity,acceleration, force, momentum Some examples of non-vectors (scalars) are: distance, speed, work, energy

3. Vector Addition When two or more vectors are added the directions must be considered. Vectors may be added Graphically or Analytically. Graphical Addition requires the use of rulers and protractors to make scale drawings of vectors tip-to-tail. (Less accurate) Analytical Addition is a strictly mathematical method using trigonometric functions (sin, cos, tan) to add the vectors together.

4. Vector addition- Graphical B A Re-draw as tip to tail by moving one of the vectors to the tip of the other. A B R R = Resultant vector which is the vector sum of A+B.The resultant always goes from the beginning (tail of first vector) to the end (tip of last vector). Given vectors A and B

5. Graphical Addition of Several Vectors- A B C Re-draw vectors tip-to-tail A B C R R = A + B + C    (direction of R) Given vectors A, B, and C Note:  represents the angle between vector B and the horizontal axis

6. Vector Subtraction To subtract vectors, add a negative vector. A negative vector has the same magnitude and the opposite direction. Example: A -A Note: A + (-A) = 0

7. Vector Subtraction A -B  Given : Vectors A and B, find R = A - B Re-draw tip to tail as A+(-B)  A R Note: In addition, vectors can be added in any order. In subtraction, you must consider the order so that the right vector is reversed when added. B

8. Equilibrant The equilibrant vector is the vector that will balance the combination of vectors given. It is always equal in magnitude and opposite in direction to the resultant vector.

9. Equilibrant Given vectors A and B, find the equilibrant A B Re-draw as tip to tail, find resultant, then draw equilibrant equal and opposite. A B R E

10. Right Triangle Trigonometry B Aadjacent opposite hypotenuse sin  = opposite hypotenuse cos  = adjacent hypotenuse tan  = opposite adjacent  And don’t forget: A 2 + B 2 = C 2 C

11. Vector Resolution A Ay AxAx  Given: vector A at angle  from horizontal. Resolve A into its components. (A x and A y ) x y Evaluate the triangle using sin and cos. cos  =A x /A so…A x = A cos  sin  = A y /A so… A y = A sin  Hint: Be sure your calculator is in degrees!

12. Vector Addition-Analytical –resolve the vectors to be added into their x- and y- components. –Add the x- components together to get a resultant vector in the x direction –Add the y- components together to get a resultant vector in the y direction –Use the pythagorean theorem to add the resultant vectors in the x- and y-components together. –Use the tan function of your resultant triangle to find the direction of the resultant. To add vectors mathematically:

13. Vector Addition-Analytical Example: Given: Vector A is 90 at 30 O and vector B is 50 at 125 O. Find the resultant R = A + B mathematically. 30 O 55 O A B A x = 90 cos 30 O = 77.9 A y = 90 sin 30 O = 45 B x = 50 cos 55 O =28.7 B y = 50 sin55 O = 41 Note: B x will be negative because it is acting along the -x axis. ByBy AxAx AyAy BxBx First, calculate the x and y components of each vector.

14. Vector Addition-Analytical (continued) Then find R: R 2 = R x 2 + R y 2 R 2 = (49.2) 2 +(86) 2, so… R = 99.1 To find direction of R:  = tan -1 ( R y / R x )  = tan -1 ( 86 / 49.2 ) = 60.2 O Find R x and R y : R x = A x + B x R y = A y + B y R x = 77.9 - 28.7 = 49.2 R y = 41 + 45 = 86 RxRx RyRy R 

15. Stating the final answer All vectors must be stated with a magnitude and direction. Angles must be specified according to compass directions( i.e. N of E) or adjusted to be measured from the +x-axis(0°). The calculator will always give the angle measured from the closest horizontal axis. CCW angles are +, CW are - * * ccw = counter- clockwise cw = clockwise