1. VECTORS ARE QUANTITIES THAT HAVE MAGNITUDE AND DIRECTION

2. Vector vs. Scalar
Scalar- Quantities that have magnitude (size) only, like mass, time, distance, energy, volume, and speed.
Vector- Quantities that have both magnitude and direction like displacement, velocity, acceleration, force, momentum, and fields.

3. DRAWING A VECTOR? Tip Tail A vector has both size and direction.
A vector is represented on paper by an arrow drawn to scale and pointing in the direction of the action
Tail

4. 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.

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

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

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

8. VECTOR PROPERTIES Commutative Property: A+B = B+A
Associative Property: (A+B)+C = A+(B+C)
Zero Property: A+(-B) = 0, if and only if,
A is equal in magnitude to B and pointing in the opposite direction.
• Subtraction: A - B = A + (-B)
• Multiplication: 3 x A = 3A

9. Vector angle ranges N E W S     NW quad 270o <  < 360o
NE quad
0 <  < 90o E W
SW quad
180o <  < 270o 
SE quad
90o <  < 180o  S

11. ADDITION OF VECTORS
3 Methods
Parallelogram Method- For a quick assessment. Good for concurrent forces.
Tip-to-Tail Method- Drawing vectors to scale on paper to find an answer. Use of a pencil, ruler and protractor needed. Good for displacements.
Mathematical Method- Determining an answer using trigonometry. The vectors need to be at right angles to one another.

12. PARALLELOGRAM METHOD
Arrange the vectors tail to tail in the correct direction and draw to scale.
Draw two identical vectors as the originals to form a parallelogram.
Draw in the diagonal of the parallelogram. This is your answer called a resultant.
Measure the resultant and find the angle.

13. THE PARALLELOGRAM METHODB B A
THE PARALLELOGRAM METHOD

14. Concurrent Forces

15. 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 canceled out by adding it to its inverse, which is called the equilibrant.

16. TIP-TO-TAIL METHOD
Arrange the scaled vectors from the tip of one to the tail of the next.
Draw the resultant from the tail of the first vector to the tip of the last vector.
Determine the magnitude of the resultant, and find the angle from the base of the resultant. Use a ruler and protractor.

17. Displacements as Vectors
Direction
Magnitude

18. A Scale and Ruler

19. The Protractor
The obtuse angle
The acute angle

20. TIP-TO-TAIL METHOD A: 12 meters 20o East of North C B A R
B: 15 meters East
Find the Resultant Displacement
C: 5 meters 30o North of West
TIP-TO-TAIL METHOD

21. MATHEMATICAL METHOD for vectors at right angles
Sketch a diagram of the vectors.
Use the pythagorean theorem to determine the magnitude of the resultant.
Use the sine, cosine, or tangent function to determine the angle from the base of the resultant.

22. TRIG REVIEW hyp opp q adj T.O.A.S.O.H.C.A.H. SOHCAHTOA
Pythagorean Theorem
hyp
opp q
adj

23. MATHEMATICAL METHOD A: 85 N, West B: 150 N, North B R R B
Find the Resultant A

24. VECTOR COMPONENTS Every vector has 2 components.
One component is horizontal, or
x-direction.
One component is vertical, or y-direction.
The 2 components are always  to each other.
Use trig functions to find them.

25. Any Old Vector
Y-Component
X-Component

26. HORIZONTAL COMPONENT
VERTICAL COMPONENT

27. Inclined Planes Ff FN
mgcosq q mg
mgsinq q

28. IN SUMMARY Vectors have magnitude and direction
3 methods to add vectors
Parallelogram
Tip-to-Tail
Mathematical
Components are perpendicular vectors to any old vector.