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8-7 Vectors

You used trigonometry to find side lengths and angle measures of right triangles. 8-7 Vectors. Perform vector operations geometrically. Perform vector operations on the coordinate plane. Definition. B. Terminal point or tip. A. Initial point or tail.

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8-7 Vectors

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  1. You used trigonometry to find side lengths and angle measures of right triangles. 8-7 Vectors • Perform vector operations geometrically. • Perform vector operations on the coordinate plane.

  2. Definition B Terminal point or tip A Initial point or tail A vector can be represented as a “directed” line segment, useful in describing paths. A vector looks like a ray, but it is NOT!! A vector has both direction and magnitude (length).

  3. Direction and Length From the school entrance, I went three blocks north. The distance (magnitude) is: Three blocks The direction is: North

  4. Direction and Magnitude The magnitude of AB is the distance between A and B. The direction of a vector is measured counterclockwise from the horizonal (positive x-axis).

  5. B B N 60° A 45° E W A S

  6. Drawing Vectors Draw vector YZ with direction of 45° and length of 10 cm. Z 10 cm • Draw a horizontal dotted line • Use a protractor to draw 45° • Use a ruler to draw 10 cm • Label the points 45° Y

  7. A. Use a ruler and a protractor to draw each vector. Include a scale on each diagram. = 80 meters at 24° west of north Using a scale of 1 cm : 50 m, draw and label an 80 ÷ 50 or 1.6-centimeter arrow 24º west of the north-south line on the north side. Answer:

  8. B. Use a ruler and a protractor to draw each vector. Include a scale on each diagram. = 16 yards per second at 165° to the horizontal Using a scale of 1 cm : 8 yd/s, draw and label a 16 ÷ 8 or 2-centimeter arrow at a 165º angle to the horizontal. Answer:

  9. Using a ruler and a protractor, draw a vector to represent feet per second 25 east of north. Include a scale on your diagram. A. B. C. D.

  10. Resultant = Vector Sum A path or trip that consists of several segments can be modeled by a sequence of vectors. The endpoint of one vector is the origin of the next vector in the chain. The figure shows a ship’s path from point M to point N that consists of five vectors. U T S V N M

  11. What is the shortest path from M to N? Write the vector sum for the boat’s trip starting with MS Resultant (Vector Sum) U T S V N M

  12. For resultants (vector sums), the following is true: XY + YZ = XZ Y Z X

  13. p. 601

  14. Types of Vectors • Parallel vectors have the same or opposite direction but not necessarily the same magnitude (length) • Opposite vectors have the same magnitude but opposite direction. • Equivalent vectors have the same magnitude and direction.

  15. Copy the vectors. Then find Step 1 , and translate it so that its tail touches the tail of . b a a a –b –b Find the Resultant of Two Vectors Subtracting a vector is equivalent to adding its opposite. Method 1 Use the parallelogram method.

  16. a – b a –b Step 2 Complete the parallelogram. Then draw the diagonal.

  17. Step 2 Draw the resultant vector from the tail of to the tip of – . –b Step 1 , and translate it so that its tail touches the tail of . –b a a – b a a – b Method 2 Use the triangle method. Answer:

  18. Copy the vectors. Then find a b A. B. C.D. a – b a – b a – b a – b

  19. Write the component form of . Vectors on the Coordinate Plane Find the change of x-values and the corresponding change in y-values. Component form of vector Simplify.

  20. Write the component form of . A. B. C. D.

  21. Assignment Page 605, 12-26 even

  22. 8-7 Vectors day 2 You used trigonometry to find side lengths and angle measures of right triangles. • Perform vector operations geometrically. • Perform vector operations on the coordinate plane.

  23. Find the magnitude and direction of Distance Formula (x1, y1) = (0, 0) and (x2, y2) = (7, –5) Find the Magnitude and Direction of a Vector Step 1 Use the Distance Formula to find the vector’s magnitude. Simplify. Use a calculator.

  24. Definition of inverse tangent Use a calculator. Answer: Step 2 Use trigonometry to find the vector’s direction.

  25. Find the magnitude and direction of A. 4; 45° B. 5.7; 45° C. 5.7; 225° D. 8; 135°

  26. p. 603 Scalar – a constant multiplied by a vector Scalar multiplication – multiplication of a vector by a scalar (dilation)

  27. Find each of the following for and . Check your answers graphically.A. Check Graphically Solve Algebraically

  28. Find each of the following for and . Check your answers graphically.B. Check Graphically Solve Algebraically

  29. Find each of the following for and . Check your answers graphically.C. Check Graphically Solve Algebraically

  30. A. B. C. D.

  31. Draw a diagram. Let represent the resultant vector. CANOEING Suppose a person is canoeing due east across a river at 4 miles per hour. If the river is flowing south at 3 miles per hour, what is the resultant speed and direction of the canoe? The component form of the vector representing the velocity of the canoe is 4, 0, and the component form of the vector representing the velocity of the river is 0, –3. The resultant vector is 4, 0 + 0, –3 or 4, –3, which represents the resultant velocity of the canoe. Its magnitude represents the resultant speed.

  32. Use the Distance Formula to find the resultant speed. Distance Formula (x1, y1) = (0, 0) and (x2, y2) = (4, –3) The resultant speed of the canoe is 5 miles per hour. Use trigonometry to find the resultant direction. Definition of inverse tangent Use a calculator. The resultant direction of the canoe is about 36.9° south of due east. Answer: Therefore, the resultant speed of the canoe is 5 mile per hour at an angle of about 90° – 36.9° or 53.1° east of south.

  33. 8-7 Assignment • Page 605, 29-33 odd, 35-40 all

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