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Mastering Vectors: Addition, Subtraction, and Components

Learn the fundamentals of vectors, addition, subtraction, and components, including scalar multiplication and unit vectors in physics.

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Mastering Vectors: Addition, Subtraction, and Components

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  1. Chapter 3 Vectors

  2. Vectors • Vectors – physical quantities having both magnitude and direction • Vectors are labeled either a or • Vector magnitude is labeled either |a| or a • Two (or more) vectors having the same magnitude and direction are identical

  3. R = A + B B A • Vector sum (resultant vector) • Not the same as algebraic sum • Triangle method of finding the resultant: • Draw the vectors “head-to-tail” • The resultant is drawn from the tail of A to the head of B

  4. Addition of more than two vectors • When you have many vectors, just keep repeating the process until all are included • The resultant is still drawn from the tail of the first vector to the head of the last vector

  5. Commutative law of vector addition A + B = B + A

  6. Associative law of vector addition (A + B) + C = A + (B + C)

  7. Negative vectors Vector (- b) has the same magnitude as b but opposite direction

  8. Vector subtraction Special case of vector addition: A - B = A + (- B)

  9. Multiplying a vector by a scalar • The result of the multiplication is a vector • c A = B • Vector magnitude of the product is multiplied by the scalar • |c| |A| = |B| • If the scalar is positive (negative), the direction of the result is the same as (opposite to that) of the original vector

  10. Vector components • Component of a vector is the projection of the vector on an axis • To find the projection – drop perpendicular lines to the axis from both ends of the vector – resolving the vector

  11. Vector components

  12. Unit vectors • Unit vector: • Has a magnitude of 1 (unity) • Lacks both dimension and unit • Specifies a direction • Unit vectors in a right-handed coordinate system

  13. Adding vectors by components In 2D case:

  14. Chapter 3 Problem 33 Vector B has z, y, and z components of 4.00, 6.00, and 3.00 units, respectively. Calculate the magnitude of B and the angles B makes with the coordinate axes.

  15. Answers to the even-numbered problems Chapter 3 Problem 4: y = 1.15; r = 2.31

  16. Answers to the even-numbered problems Chapter 3 Problem 6: 310 km at 57° S of W

  17. Answers to the even-numbered problems Chapter 3 Problem 16: 1.31 km north; 2.81 km east

  18. Answers to the even-numbered problems Chapter 3 Problem 20: - 25.0 m i^ + 43.3 m j^

  19. Answers to the even-numbered problems Chapter 3 Problem 24: (b) 5.00 i^ + 4.00 j^, 6.40 at 38.7°; – 1.00 i^ + 8.00 j^, 8.06 at 97.2°

  20. Answers to the even-numbered problems Chapter 3 Problem 30: C = 7.30 cm i^ - 7.20 cm j^

  21. Answers to the even-numbered problems Chapter 3 Problem 52: (a) 2.00, 1.00, 3.00 (b) 3.74 (c) θx = 57.7°, θy = 74.5°, θz = 36.7°

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