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

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

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

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

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

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

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

Determine the Resultant Vector B R A A B B Order doesn’t matter when adding vectors!

Determine the Resultant Vector B A R B

Determine the Resultant Vector B C A B R C

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

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

Determine the Resultant Vector B C 30N @ 45° A B 30N @ 225° C

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)

Calculate the Resultant Vector Horizontal Component Vertical Component A 0N +10N B R 10N @ 90° 14.14N @ 45° 10N @ 0°

Calculate the Resultant Vector Horizontal Component Vertical Component A -7.66N -6.43N B -10N 0N R -17.66N 10N @ 180° 10N @ 40° below –x-axis 18.79N @ 20° below –x-axis

Calculate the Resultant Vector Horizontal Component Vertical Component A -7.66N -6.43N B +8.66N 5N R +1N -1.43N 10N @ 30° 10N @ 40° below –x-axis 1.74N @ 55° below +x-axis

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

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

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; 10N – 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

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?