1. Vectors
An Introduction

2. There are two kinds of quantities…
Vectors are quantities that have both magnitude and direction (e.g., displacement, velocity, acceleration).
Scalars are quantities that have magnitude only (e.g., position, speed, time, mass).

3. Notating vectors Vector: R R R This is how you draw a vector. head
This is how you draw a vector. R
head
tail

4. Notating scalars
Scalar: R
There is no standard way to draw a scalar!

5. Direction of Vectors A  x  x B

6. Vector angle ranges     II 90o <  < 180o Ix y II
90o <  < 180o   I
0 <  < 90o
III
180o <  < 270o  IV
270o <  < 360o 

7. Magnitude of Vectors
The best way to describe the magnitude of a vector is to measure the length of the vector.
The length of the vector is proportional to the magnitude of the quantity it represents.

8. Magnitude of Vectors A B
If vector A represents a displacement of three miles to the north… B
Then vector B, which is twice as long, would represent a displacement of six miles to the north!

9. Equal Vectors
Equal vectors have the same length and direction, and represent the same quantity (such as force or velocity).

10. Inverse Vectors
Inverse vectors have the same length, but opposite direction. A -A

11. Vectors: x-component
Ax = A cos  Ax A  x

12. Vectors: y-component
Ay = A sin  A  x Ay

13. Vectors: angle y
 = tan-1 (Ry/Rx) Ry  x Rx

14. Vectors: magnitude y R
R = √ (Rx2 + Ry2) Ry x Rx

15. Graphical Addition of VectorsB A R
A + B = R
R is called the resultant vector!

16. The Resultant and the Equilibrant
The sum of two or more vectors is called the resultant vector.
The resultant vector can replace the vectors from which it is derived.
The resultant is completely cancelled out by adding it to its inverse, which is called the equilibrant.

17. Graphical Addition of VectorsB A E R
A + B = R
E is called the equilibrant vector!

18. Component Addition of Vectors
Resolve each vector into its x- and y-components.
Ax = Acos Ay = Asin
Bx = Bcos By = Bsin
Cx = Ccos Cy = Csin etc.
Add the x-components (Ax, Bx, etc.) together to get Rx and the y-components (Ay, By, etc.) to get Ry.

19. Component Addition of Vectors
Calculate the magnitude of the resultant with the Pythagorean Theorem (R = Rx2 + Ry2).
Determine the angle with the equation  = tan-1 Ry/Rx.

20. Relative Motion Vs Vw
Vt = Vs + Vw Vw

21. Relative Motion Vs
Vt = Vs + Vw Vw Vw

22. Relative Motion Vw Vs
Vt = Vs + Vw Vw