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Introduction to Vectors in Physics: Scalars, Vectors, and Trigonometry

Explore the fundamentals of vectors in physics, learning about scalars, vectors, graphical addition, subtraction, and multiplication by a scalar. Discover how to find components of vectors, use trigonometry for calculations, and work with unit vectors. Practice drawing vectors, adding multiple vectors, and understanding the dot product.

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Introduction to Vectors in Physics: Scalars, Vectors, and Trigonometry

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  1. Physics Chapter 1Fall 2012 Vectors and Right Triangle Trigonometry

  2. Vectors and scalars • A scalar quantity can be described by a single number. • A vector quantity has both a magnitude and a direction in space. • In this book, a vector quantity is represented in boldface italic type with an arrow over it: A. • The magnitude of A is written as A or |A|.   

  3. Drawing vectors • Draw a vector as a line with an arrowhead at its tip. • The length of the line shows the vector’s magnitude. • The direction of the line shows the vector’s direction. • Figure 1.10 shows equal-magnitude vectors having the same direction and opposite directions.

  4. Adding two vectors graphically • Two vectors may be added graphically using either the parallelogrammethod or the head-to-tail method.

  5. Adding more than two vectors graphically • To add several vectors, use the head-to-tail method. • The vectors can be added in any order.

  6. Q1.2 Consider the vectors shown. Which is a correct statement about A. x-component > 0, y-component > 0 B. x-component > 0, y-component < 0 C. x-component < 0, y-component > 0 D. x-component < 0, y-component < 0 E. x-component = 0, y-component > 0

  7. A1.2 Consider the vectors shown. Which is a correct statement about A. x-component > 0, y-component > 0 B. x-component > 0, y-component < 0 C. x-component < 0, y-component > 0 D. x-component < 0, y-component < 0 E. x-component = 0, y-component > 0

  8. Subtracting vectors • This shows us how to subtract vectors.

  9. Q1.3 Consider the vectors shown. Which is a correct statement about A. x-component > 0, y-component > 0 B. x-component > 0, y-component < 0 C. x-component < 0, y-component > 0 D. x-component < 0, y-component < 0 E. x-component = 0, y-component > 0

  10. A1.3 Consider the vectors shown. Which is a correct statement about A. x-component > 0, y-component > 0 B. x-component > 0, y-component < 0 C. x-component < 0, y-component > 0 D. x-component < 0, y-component < 0 E. x-component = 0, y-component > 0

  11. Multiplying a vector by a scalar  • If c is a scalar, the product cA has magnitude |c|A. • Figure 1.15 illustrates multiplication of a vector by a positive scalar and a negative scalar.

  12. Addition of two vectors at right angles • First add the vectors graphically. • Then use trigonometry to find the magnitude and direction of the sum.

  13. Components of a vector • Adding vectors graphically provides limited accuracy. Vector components provide a general method for adding vectors. • Any vector can be represented by an x-component Ax and a y-component Ay. • Use trigonometry to find the components of a vector: Ax =Acosθ and Ay = Asinθ, where θ is measured from the +x-axis toward the +y-axis.

  14. Positive and negative components—Figure 1.18 • The components of a vector can be positive or negative numbers, as shown in the figure.

  15. Q1.6 Consider the vectors shown. What are the components of the vector A. Ex= –8.00 m, Ey = –2.00 m B. Ex= –8.00 m, Ey = +2.00 m C. Ex= –6.00 m, Ey = 0 D. Ex= –6.00 m, Ey = +2.00 m E. Ex= –10.0 m, Ey = 0

  16. A1.6 Consider the vectors shown. What are the components of the vector A. Ex= –8.00 m, Ey = –2.00 m B. Ex= –8.00 m, Ey = +2.00 m C. Ex= –6.00 m, Ey = 0 D. Ex= –6.00 m, Ey = +2.00 m E. Ex= –10.0 m, Ey = 0

  17. Finding components • We can calculate the components of a vector from its magnitude and direction.

  18. Q1.1 What are the x- and y-components of the vector A. Ex= E cos b, Ey= E sin b B. Ex= E sin b, Ey= E cos b C. Ex= –E cos b, Ey= –E sin b D. Ex= –E sin b, Ey= –E cos b E. Ex= –E cos b, Ey= E sin b

  19. A1.1 What are the x– and y–components of the vector A. Ex= E cos b, Ey= E sin b B. Ex= E sin b, Ey= E cos b C. Ex= –E cos b, Ey= –E sin b D. Ex= –E sin b, Ey= –E cos b E. Ex= –E cos b, Ey= E sin b

  20. Calculations using components • We can use the components of a vector to find its magnitude and direction: • We can use the components of a set of vectors to find the components of their sum:

  21. Adding vectors using their components

  22. Unit vectors • A unit vector has a magnitude of 1 with no units. • The unit vector î points in the +x-direction, points in the +y-direction, and points in the +z-direction. • Any vector can be expressed in terms of its components as A=Axî+ Ay + Az. 

  23. Q1.7 The angle  is measured counterclockwise from the positive x-axis as shown. For which of these vectors is  greatest? A. B. C. D.

  24. A1.7 The angle  is measured counterclockwise from the positive x-axis as shown. For which of these vectors is  greatest? A. B. C. D.

  25. The scalar product (also called the “dot product”) of two vectors is • Figures 1.25 and 1.26 illustrate the scalar product. The scalar product

  26. In terms of components, • Example 1.10 shows how to calculate a scalar product in two ways. Calculating a scalar product [Insert figure 1.27 here]

  27. Finding an angle using the scalar product • Example 1.11 shows how to use components to find the angle between two vectors.

  28. The vector product (“cross product”) of two vectors has magnitude • and the right-hand rule gives its direction. See Figures 1.29 and 1.30. The vector product—Figures 1.29–1.30

  29. Calculating the vector product—Figure 1.32 • Use ABsin to find the magnitude and the right-hand rule to find the direction. • Refer to Example 1.12.

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