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Vocabulary

vector initial point terminal point standard position direction magnitude quadrant bearing true bearing parallel vectors. equivalent vectors opposite vectors resultant triangle method parallelogram method zero vector. components rectangular components. Vocabulary.

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Vocabulary

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  1. vector • initial point • terminal point • standard position • direction • magnitude • quadrant bearing • true bearing • parallel vectors • equivalent vectors • opposite vectors • resultant • triangle method • parallelogram method • zero vector • components • rectangular components Vocabulary

  2. Utilizing your book, what is the difference between a scalar and a vector?

  3. Identify Vector Quantities A. State whether a hockey puck shot northwest at 60 miles per hour is a vector quantity or a scalar quantity. This quantity has a magnitude of 60 miles per hour and a direction of northwest. This directed distance is a vector quantity. Answer:vector Example 1

  4. Identify Vector Quantities B. State whether a tennis ball served at 110 miles per hour is a vector quantity or a scalar quantity. This quantity has a magnitude of 110 miles per hour, but no direction is given. Speed is a scalar quantity. Answer:scalar Example 1

  5. Identify Vector Quantities C. State whether a sprinter running 100 meters north is a vector quantity or a scalar quantity. This quantity has a magnitude of 100 meters and a direction of north. This directed distance is a vector quantity. Answer:vector Example 1

  6. State whether a tow truck pulling a car due east with a force of 100 newtons is a vector quantity or a scalar quantity. A. vector B. scalar Example 1

  7. V Represent a Vector Geometrically A. Use a ruler and a protractor to draw an arrow diagram for v = 10 newtons of force at 30° to the horizontal. Include a scale on the diagram. Using a scale of 1 cm : 5 N, draw and label a 10 ÷ 5 or 2-cm arrow in standard position at a 30° angle to the x-axis. Sample Answer: Example 2

  8. Represent a Vector Geometrically B. Use a ruler and a protractor to draw an arrow diagram for z = 25 meters per second at a bearing of S70°W. Include a scale on the diagram. Using a scale of 1 cm : 10 m/s, draw and label a 25 ÷ 10 or 2.5-cm arrow 70° west of south. Sample Answer: Example 2

  9. Represent a Vector Geometrically C. Use a ruler and a protractor to draw an arrow diagram for t = 10 miles per hour at a bearing of 025°. Include a scale on the diagram. Using a scale of 1 cm : 10 mi/h, draw and label a 10 ÷ 10 or 1-cm arrow at an angle of 25° clockwise from the north. Sample Answer: Example 2

  10. A. B. C. D. Use a ruler and protractor to draw an arrow diagram for n = 20 feet per second at a bearing of 080°. Include a scale on the diagram. Example 2

  11. Video

  12. Key Concept 3

  13. Find the Resultant of Two Vectors HIKING While hiking in the woods, Shelly walks 2 kilometers N30°W from her camp, and then walks 2 kilometers directly east. How far and at what quadrant bearing is Shelly from her camp? Let p = walking 2 kilometers N30°W and q = walking 2 kilometers due east. Draw a diagram to represent p and q using a scale of 1 cm : 1 km. Example 3

  14. Find the Resultant of Two Vectors Use a ruler and a protractor to draw a 2-centimeter arrow 30° west of north to represent p and a2-centimeter arrow due east to represent q. Example 3

  15. Find the Resultant of Two Vectors Method 1Triangle Method Translate q so that its tail touches the tip of p. Then draw the resultant vector p + qas shown. Example 3

  16. Find the Resultant of Two Vectors Method 2 Parallelogram Method Translate q so that its tail touches the tail of p. Then complete the parallelogram and draw the diagonal, resultant p + q, as shown. Example 3

  17. Find the Resultant of Two Vectors Both methods produce the same resultant vector p + q. Measure the length ofp + q and then measure the angle this vector makes with the north-south line as shown. The vector’s length of 2 centimeters represents 2 kilometers. Therefore, Shelly is 2 kilometers at a bearing of 30° east of north or N30°E from her starting position. Example 3

  18. Find the Resultant of Two Vectors Answer:2 km, N30°E Example 3

  19. ORIENTEERING In an orienteering competition, Miguel walks N80E for 150 feet and then walks 200 feet due east. How far and at what quadrant bearing is Miguel from his starting position? A. about 349 feet at a bearing of N86°E B. about 270 feet at a bearing of N37°E C. about 270 feet at a bearing of N47°E D. about 350 feet at a bearing of N80°E Example 3

  20. Key Concept 4

  21. Operations with Vectors Draw a vector diagram of a – 3b. Rewrite the expression as the addition of two vectors: a – 3b = a + (– 3b). To represent –3b, draw a vector 3 times as long as b in the opposite direction from b. Example 4

  22. Operations with Vectors Then use the triangle method to draw the resultant vector. Answer: Example 4

  23. A. B. C. D. Draw a vector diagram of n + 2m. Example 4

  24. Use Vectors to Solve Navigation Problems AVIATION An airplane is flying with an airspeed of 475 miles per hour on a heading of 070°. If an 80-mile-per-hour wind is blowing from a true heading of 120°, determine the velocity and direction of the plane relative to the ground. Step 1Draw a diagram to represent the heading and wind velocities. Example 5

  25. Use Vectors to Solve Navigation Problems Translate the wind vector as shown in the diagram below, and use the triangle method to obtain the resultant vector representing the plane’s ground velocity and direction g. In the triangle formed by these vectors, γ= 120° – 70° or 50°. Example 5

  26. Take the positive square root of each side. Use Vectors to Solve Navigation Problems Step 2Use the Law of Cosines to find |g|, the plane’s speed relative to the ground. c 2 = a2 + b2 – 2abcos γ Law of Cosines |g|2 = 802 + 4752 – 2(80)(475)cos 50° c = |g|, a = 80, b = 475, and γ = 50° Example 5

  27. Law of Sines c = |g| or 428.0, a = 80, and γ = 50 Use Vectors to Solve Navigation Problems ≈ 428.0 Simplify. The ground speed of the plane is about 428.0 miles per hour. Step 3The heading of the resultant g is represented by angle θ, as shown above. To find θ, first calculate α using the Law of Sines. Example 5

  28. Solve for sin α. Apply the inverse sine function. Simplify. Use Vectors to Solve Navigation Problems The measure of θ is 70° – α, which is 70° – 8.2° or 61.8°. Therefore, the speed of the plane relative to the ground is about 428.0 miles per hour at about 061.8°. Example 5

  29. Use Vectors to Solve Navigation Problems Answer:The velocity of the plane relative to the ground is about 428.0 miles per hour at a bearing of about 061.8°. Example 5

  30. ROWING Jamie rows her boat due east at a speed of 20 feet per second across a river directly toward the opposite bank. At the same time, the current of the river is carrying her due south at a rate of 4 feet per second. Find Jamie’s speed and direction relative to the shore. A. Jamie is rowing at a resultant velocity of 24 ft/s in a direction directly east. B. Jamie is rowing at a resultant velocity of 20.4 ft/s in a direction of S11.3°E. C. Jamie is rowing at a resultant velocity of 20.4 ft/s in a direction of S29.8°E. D. Jamie is rowing at a resultant velocity of 20.4 ft/s in a direction of S78.7°E. Example 5

  31. Resolve a Force into Rectangular Components A. GARDENING While digging in his garden, Will pushes a shovel into the ground with a force of 630 newtons at an angle of 70° with the ground. Draw a diagram that shows the resolution of the force that Will exerts into its rectangular components. Example 6

  32. Resolve a Force into Rectangular Components Will’s push can be resolved into a horizontal push x forward and a vertical push y downward as shown. Answer: Example 6

  33. Resolve a Force into Rectangular Components B. GARDENING While digging in his garden, Will pushes a shovel into the ground with a force of 630 newtons at an angle of 70° with the ground. Find the magnitudes of the horizontal and vertical components of the force. The horizontal and vertical components of the force form a right triangle. Use the sine or cosine ratios to find the magnitude of each force. Example 6

  34. Right triangle definitions of cosine and sine Solve for x and y. Use a calculator. Resolve a Force into Rectangular Components Answer:horizontal component ≈ 215.47 N; vertical component ≈592.01 N Example 6

  35. SOCCER A player kicks a soccer ball so that it leaves the ground with a velocity of 39 feet per second at an angle of 37° with the ground. Find the magnitude of the horizontal and vertical components of the velocity. A. horizontal component ≈ 28.75 ft/s; vertical component ≈ 23.28 ft/s B. horizontal component ≈ 37 ft/s; vertical component ≈ 39 ft/s C. horizontal component ≈ 31.15 ft/s; vertical component ≈ 23.47 ft/s D. horizontal component ≈ 23.47 ft/s; vertical component ≈ 31.15 ft/s Example 6

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