Unit 5: Vectors - doctortang.com

1 Applied Math 30 unit 5: Vectors unit 5: Vectors 7-1: Vectors and Scalars Scalar Quantity: - quantity that involves a Magnitude of measurement but NO Direction. vector Quantity: - quantity that involves a Magnitude of measurement AND a Direction. Scalar Quantities Examples vector Quantities Examples Distance 20 m Displacement 20 m [North]. (Length WITHOUT Direction) (Length WITH Direction). Speed 50 km/h Velocity 50 km/h [280o]. (How Fast an object moves WITHOUT Direction) (How Fast an object moves WITH Direction). Mass 44 kg Force 40 N downward (How Much Stuff is IN an object) (Mass Acceleration).

2 Energy 500 kJ Weight 500 N (always downward). (Emits in ALL Directions) (Force due to Gravity). Temperature 25 oC Friction 15 N (against the direction of motion). (Average Kinetic Energy of an object) (Resistance Force due to Surface Conditions). Time 65 minutes Acceleration 5 m/s2 [NW]. (How Fast Velocity Changes over Time). vector Notation: - a method of indicating that the quantity is a vector by placing an arrow on top of the variable. Bearing: - compass bearing STARTS at the North (0o) and rotates CLOCKWISE. N = 0o Example: vector AB or v = 30 km/h [NE] or N45oE.

3 NW = 315 o NE = 45 o (From N, move 45o towards E). B Ending Point (Head). W = 270o E = 90o Magnitude v = 30 km/h SW = 225o SE = 135o S = 180o A Starting Point (Tail). Copyrighted by Gabriel tang , Page 109. unit 5: Vectors Applied Math 30. Draw Vectors with Proper Scale Example 1: Draw F1 = 60 N [115o] Example 2: Draw F2 = 45 N [240o]. (Use 1 cm = 10 N) (Use 1 cm = 10 N). N = 0o N = 0o 115o F2 = 45 N 240o F1 = 60 N. cm 6 cm Example 3: Draw d1 = 32 km [N20oW] Example 4: Draw d 2 = 56 m [E35No]. (Use 1 cm = 10 km) (Use 1 cm = 10 m). N N. d 1 = 32 km 20o d 2 = 56 m cm cm o 55.

4 70o 35o W E. OR we can say d 1 = 32 km [W70oN]. OR we can say d 2 = 56 m [N55oE]. Equal and Opposite Vectors Equal Vectors : - Vectors that have the SAME Magnitude AND Direction. B. D. AB = CD where AB = CD and they have the SAME Direction. A. C. Page 110. Copyrighted by Gabriel tang , Applied Math 30 unit 5: Vectors Opposite Vectors : - Vectors that have the SAME Magnitude but DIFFERENT Direction. B. C. AB = CD where AB = CD and they have OPPOSITE Direction. A. D. Example 5: Draw the opposite velocity vector to v = 40 km/h [S63Wo] (Use 1 cm = 10 km/h). N v = 40 km/h 63o 4 cm W E.

5 V = 40 km/h 63o Opposite Velocity vector = v = 40 km/h [N63oE]. 4 cm S. Example 6: Draw the opposite acceleration vector to a = 55 m/s2 [115o] (Use 1 cm = 10 m/s2). N. a = 55 m/s2. 115o cm Opposite Acceleration vector = a = 55 m/s2 [295o]. W E. 180o + 115o = 295o a = 55 m/s2. cm S. 7-1 Assignment: pg. 307 309 #1 to 9. Copyrighted by Gabriel tang , Page 111. unit 5: Vectors Applied Math 30. 7-2: Adding Vectors Using Scale Diagrams Resultant vector : - the vector that is the result of vector addition or subtraction. - from the Starting Point of the First vector to the Ending Point of the Last vector .

6 Adding Vectors (always Connect Vectors FROM HEAD TO TAIL). Slide vector v along vector u u+v = R u until the Vectors are lined up u v From HEAD to TAIL. R. v Example 1: Draw AB + CD. B CD. C D. AB B. A AB. A R. C. CD D. Subtracting Vectors (ADD OPPOSITE Vectors ). u v = u+ v = R ( ) v Create an Opposite Add From u u HEAD To TAIL R. vector v ( v ). u v v v Example 2: Draw AB CD. B. B. AB D CD. AB C. B. A. A D AB. CD C R. C CD D. CD D A. AB + ( CD ) = R. Page 112. Copyrighted by Gabriel tang , Applied Math 30 unit 5: Vectors Example 3: A ship left the dock and traveled north at 50 km/h for 2 hours, then it turned west at 60 km/h for 3 hour.

7 A. What is the net resultant displacement vector , R ? (Use 1 cm = 20 km). d 1 = 50 km/h [N] 2 hr d 1 = 100 km [North] (5 cm). d 2 = 60 km/h [N] 3 hr d 2 = 180 km [West] (9 cm). d 2 = 180 km [West] (9 cm). measured ( cm). R = cm 20 km/cm measured d 1 = 100 km [North] (5 cm). R = 206 km angle (61o). R = 206 km [N61oW] or [W29oN] or [299o] Bearing 360o 61o = 299o b. What is the displacement vector the ship must follow to return to the dock? N. Bearing 299o 180o (or 90o + 29o) = 119o E. 29o Return Displacement vector = Opposite Resultant vector S. R = 206 km R = 206 km [S61oE] or [E29oS] or [119o].

8 Example 4: A plane is flying south at 400 km/h and a steady wind is blowing from the east at 100 km/h. If a sudden gust of wind appears from the south at 150 km/h. What is the resultant velocity vector of the plane? (Use 1 cm = 100 km/h) N. v 2 = 100 km/h [West]. (from the east) (1 cm). Bearing 180o + 22o = 202o measured angle (22o). v 1 = 400 km/h [South] (4 cm) R = cm 100 km/h / cm v 1 + v .3 = 250 km/h [South] ( cm). R = 270 km/h v 2 = 100 km/h [West]. v 3 = 150 km/h [North]. (from the south) ( cm). R = 270 km/h [S22oW] or [W68oS] or [202o]. Copyrighted by Gabriel tang , Page 113.

9 unit 5: Vectors Applied Math 30. Example 5: A kayak left Port Alberni due west for 25 km. It then turned at a bearing of 300o and traveled on for 40 km. Hearing the sighting of killer whales, it turned at a bearing of 40o for 10 km. What is the kayak's net displacement from Port Alberni? (Use 1 cm = 5 km). measured ( cm) R = cm 5 km/cm Bearing 40o d 3 = 10 km R = 60 km [40o] (2 cm). R = 60 km [W28oN] or [N62oW] or [298o]. d 2 = 40 km [300o] (8 cm). measured angle (28o). Bearing 300o d 1 = 25 km [West] Bearing 270o + 28o = 298o (5 cm). Equilibrant: - the opposite resultant force of an object which does not move, but is acted on by other forces.

10 Example 6: Two tow trucks are trying to pull a heavy trailer out of a ditch. One tow truck is applying a force of 4500 N at N40oW and the other truck is pulling with a force of 6500 N at 210o. The trailer remained stuck in the ditch. a. Calculate the resultant force on the heavy trailer. (Use 1 cm = 1000 N). F1 = 4500 N [N40oW] ( cm). N40oW N. F2 = 6500 N [210o]. Bearing 210o measured angle (20o). Bearing 270o 20o = 250o measured ( cm). R = cm 1000 N/cm R = 6600 N F2 = 6500 N [210o] ( cm). R = 6600 N [W20oS] or [S70oW] or [250o]. Page 114. Copyrighted by Gabriel tang , Applied Math 30 unit 5: Vectors b.