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Chapter 3

Chapter 3. Vectors in Physics (Continued). Outline. Components of a vector How to find the components of a vector if knowing its magnitude and direction How to find the magnitude and direction of a vector if knowing its components Express a vector in terms of unit vectors

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Chapter 3

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  1. Chapter 3 Vectors in Physics (Continued) PHY 1151 Principles of Physics I

  2. Outline • Components of a vector • How to find the components of a vector if knowing its magnitude and direction • How to find the magnitude and direction of a vector if knowing its components • Express a vector in terms of unit vectors • Adding vectors using the Components Method PHY 1151 Principles of Physics I

  3. Components Method for Adding Vectors • The graphical method of adding vectors is not recommended when high accuracy is required or in three-dimensional problems. • Components method (rectangular resolution): A method of adding vectors that uses the projections of vectors along coordinate axes. PHY 1151 Principles of Physics I

  4. Components of a Vector • Components of a vector: The projections of a vector along coordinate axes are called the components of the vector. • Vector A and its components Ax and Ay • The component Ax represents the projection of A along the x axis. • The component Ay represents the projection of A along the y axis. PHY 1151 Principles of Physics I

  5. Find the Components of a Vector Given its Magnitude and Direction • If vector A has magnitude A and direction , then its components are • Ax = A cos • Ay = A sin • Note: According to convention, angle  is measured counterclockwise from the +x axis. PHY 1151 Principles of Physics I

  6. y II I x III IV Signs of the Components Ax and Ay • The signs of the components Ax and Ay depend on the angle , or in which quadrants vector A lies. • Component Ax is positive if vector Ax points in the +x direction. • Component Ax is negative if vector Ax points in the -x direction. • The same is true for component Ay. PHY 1151 Principles of Physics I

  7. Example: Find the Components of a Vector • Find Ax and Ay for the vector A with magnitude and direction given by • (1) A = 3.5 m and  = 60°. • (2) A = 3.5 m and  = 120°. • (3) A = 3.5 m and  = 240°. • (4) A = 3.5 m and  = 300°. PHY 1151 Principles of Physics I

  8. Find the Magnitude and Direction of A Given its Components Ax and Ay • The magnitude and direction of A are related to its components through the expressions: • A= (Ax2 + Ay2)1/2 •  = tan-1(Ay/Ax) • Note: Pay attention to the signs of Ax and Ay to find the correct values for . PHY 1151 Principles of Physics I

  9. Example: Find the Magnitude and Direction of a Vector • Find magnitude B and direction  for the vector B with components • (1) Bx = 75.5 m and By = 6.20 m. • (2) Bx = -75.5 m and By = 6.20 m. • (3) Bx = -75.5 m and By = -6.20 m. • (4) Bx = +75.5 m and By = -6.20 m. PHY 1151 Principles of Physics I

  10. Express Vectors Using Unit Vectors • Unit vectors: A unit vector is a dimensionless vector having a magnitude of exactly 1. • Unit vectors are used to specify a given direction and have no other physical significance. • Symbols i, j, and k represent unit vectors pointing in the +x, +y, and +z directions. • Using unit vectors i and j, vector A is expressed as: A = Axi + Ayj PHY 1151 Principles of Physics I

  11. Adding Vectors Using the Components Method • Suppose that A = Axi + Ayj and B = Bxi + Byj. • Then, the resultant vector R = A + B = (Ax + Bx)i + (Ay + By)j. • When using the components method to add vectors, all we do is find the x and y components of each vector and then add the x and y components separately. PHY 1151 Principles of Physics I

  12. Example: The Sum of Two Vectors (with Components Method) • Two vectors A and B lie in the xy plane and are given by A = (2.0i + 2.0j) m and B = (2.0i - 4.0j) m. • (1) Find the sum of A and B expressed in terms of unit vectors. • (2) Find the x and y components of the sum. • (3) Find the magnitude R and direction  of the the sum. PHY 1151 Principles of Physics I

  13. Example: Adding Vectors Using Components • A commuter airplane takes a route shown in the figure. First, it flies from the origin of the coordinate system shown to city A, located 175 km in a direction 30.0° north of east. Next, it flies 153 km 20.0° west of north to city B. Finally, it flies 195 km due west to city C. • Find the location of city C relative to the origin. o PHY 1151 Principles of Physics I

  14. Homework • Chapter 3, Page 73, Problems: #4, 8, 14, 21, 26. PHY 1151 Principles of Physics I

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