1. Aim: What other details can be discussed about vectors?
Oh Yeah!
Vectors
Aim: What other details can be discussed about vectors?

2. How to Read Angles
Standard Position – The angle of COUNTERCLOCKWISE rotation starting from 0° to the vector (0° - 360°)
Reference Angle – The ACUTE angle between the vector and the (+ or -) x-axis
Compass Angle – [Angle] [Rotation] of [Start]
“30° North of East”
Rotation Direction
Starting Direction

3. Example #1 90° N 0° 180° E W 270° S Standard 30° Reference 30° above
the +x-axis
Compass 1
North of East
Compass 2
60°
East of North
60°
30° 0° E
180° W
270° S

4. Example #2 90° N 0° 180° E W 270° S Standard 115° Reference 65° above
the –x-axis
Compass 1
25°
West of North
Compass 2
65°
North of West
25°
65°
115° 0° E
180° W
270° S

5. Example #3 90° N 0° 180° E W 270° S Standard 270° Reference 90° below
the x-axis
Compass 1
South
Compass 2 0° E
180° W
270° S

6. Example #4 90° N 0° 180° E W 270° S Standard 315° Reference 45° below
the +x-axis
Compass 1
45°
South of East
Compass 2
East of South
315°
180° W
45° 0° E
45°
270° S

7. Adding Vectors Tip-to-Tail
When adding two or more vectors tip-to-tail you would typically REDRAW a vector so the TAIL of the NEXT vector connects to the TIP of the PREVIOUS vector. But there are some important rules to keep in mind
The vector’s length (magnitude) STAYS THE SAME
The vector’s orientation (direction) STAYS THE SAME
The RESULTANT VECTOR is the only vector that is drawn from the TAIL of the FIRST VECTOR to the TIP of the LAST VECTOR

8. Determine the Resultant VectorB R A A B B
Order doesn’t matter when adding vectors!

9. Determine the Resultant VectorB A R B

10. Determine the Resultant VectorB C A B R C

11. Equilibrant
A special vector that is EQUAL in MAGNITUDE but faces the OPPOSITE DIRECTION (180°) of the resultant. Causes the system to reach EQUILIBRIUM, where the sum of all vectors is ZERO. Cancels out the resultant

12. Determine the Equilibrant Vector
20N, NE
20N, SW A B

13. Determine the Resultant VectorB C
45° A B
225° C

14. Adding Vectors Algebraically
Determine the Horizontal and Vertical components of all the vectors
Add/Subtract ALL Horizontal Values (Direction Matters)
This will be the RESULTANT HORIZONTAL component
Add/Subtract ALL Vertical Values (Direction Matters)
This will be the RESULTANT VERTICAL component
Use Pythagorean Theorem to calculate the resultant MAGNITUDE
Use Trig to determine the resultant ANGLE (Direction)

15. Calculate the Resultant Vector
Horizontal Component
Vertical Component A 0N
+10N B R
90°
45° 0°

16. Calculate the Resultant Vector
Horizontal Component
Vertical Component A
-7.66N
-6.43N B
-10N 0N R
-17.66N
180°
40°
below –x-axis
20° below –x-axis

17. Calculate the Resultant Vector
Horizontal Component
Vertical Component A
-7.66N
-6.43N B
+8.66N 5N R
+1N
-1.43N
30°
40°
below –x-axis
55° below +x-axis

18. Maximum & Minimum Resultants
Calculating the MAXIMUM resultant requires two vectors to face the SAME DIRECTION. The angle between them must be 0°. (ADDITION)
Calculating the MINIMUM resultant requires two vectors to face OPPOSITE DIRECTIONS. The angle between them must 180°. (SUBTRACTION)
Max. Resultant
180°
Min. Resultant

19. Possible Resultants Min = 3N  Max = 17N
Possible resultants are in the RANGE between the minimum and maximum resultants. 7N
10N
Min = 3N 
Max = 17N

20. Examples
Determine the minimum and maximum resultant between a 3N force and a 5N force
2N – 8N
Is it possible to have a resultant force of 25N when a 5N force and a 15N force act concurrently on an object?
No; N – 20N
Is it possible to have a resultant force of 15N when a 5N force and a 15N force act concurrently on an object?
Yes; 10N – 20N

21. SUMMARY Describe the “Tip-to-Tail” Method of adding vectors
Describe the Algebraic Method of adding vectors
At what angle between two vectors will give you the maximum resultant?
At what angle between two vectors will give you the minimum resultant?