ANNOUNCEMENT: THE OFFICE OF ACCESSIBILITY RESOURCE CENTER IS LOOKING FOR A STUDENT IN THIS CLASS TO VOLUNTEER TO PROVIDE NOTES FOR THIS CLASS. THE STUDENT WILL BE PAID A STIPEND FOR THE SEMESTER. INTERESTED STUDENT SHOULD COME BY OUR OFFICE AT 2021 MESA VISTA HALL TO COMPLETE THE REQUIRED HIRING PAPERWORK. Unregistered iClickers (bring your clicker and see me to claim) Vector Addition and Components Lecture #3 - PHYC 160 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. Copyright © 2012 Pearson Education Inc. In 3-dimensions A A Axiˆ Ay ˆj Ax 2 Az kˆ Ay 2 Az 2 Unit vectors—Figures 1.23–1.24 • A unit vector has a magnitude of 1 with no units. • The unit vector î points in the $ +x-direction, $jj points in the +y$k points in the direction, and $ k +z-direction. • Any vector can be expressed in terms of its components as → $k. A =Axî+ Ay $$jj + Az $ k • Follow Example 1.9. Copyright © 2012 Pearson Education Inc. Q1.7 The angle θ is measured counterclockwise from the positive x-axis as shown. For which of these vectors is θ greatest? A. 24 iˆ +18 ˆj B. −24 iˆ − 18 ˆj ˆ ˆ C. −18 i + 24 j D. −18 iˆ − 24 ˆj © 2012 Pearson Education, Inc. Adding more than two vectors graphically—Figure 1.13 • To add several vectors, use the head-to-tail method. • The vectors can be added in any order. Subtracting vectors • Figure 1.14 shows how to subtract vectors. Addition of two vectors at right angles • First add the vectors graphically. • Then use trigonometry to find the magnitude and direction of the sum. • Follow Example 1.5. You didnt know this, but you are adding vectors when you give directions on a map Copyright © 2012 Pearson Education Inc. You CPS Question 3-1 • Which vector most closely represents A B with A and ? B A. B. C. D. Components of a vector—Figure 1.17 • 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 ycomponent Ay. • Use trigonometry to find the components of a vector: Ax = Acos θ and Ay = Asin θ, where θ is measured from the +x-axis toward the +y-axis. Copyright © 2012 Pearson Education Inc. Calculations using components • We can use the components of a vector to find its magnitude A A = Ax2 + Ay2 and tanθ = y and direction: A x • We can use the components of a set of vectors to find the components of their sum: Rx = Ax + Bx + Cx +L , Ry = Ay + By +Cy +L • Refer to Problem-Solving Strategy 1.3. Copyright © 2012 Pearson Education Inc. Q1.2 Consider the vectors shown. Which is a correct statement about ! ! A+ B? 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 © 2012 Pearson Education, Inc. Q1.3 Consider the vectors shown. Which is a correct statement about ! ! A − B? 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 © 2012 Pearson Education, Inc. Q1.5 Which ! !of the following statements is correct for any two vectors A and B ? ! ! A. The magnitude of A − B ! ! B. The magnitude of A − B ! ! C. The magnitude of A − B ! ! D. The magnitude of A − B ! ! E. The magnitude of A − B © 2012 Pearson Education, Inc. is A – B. is A + B. is greater than or equal to |A – B|. ! ! is less than the magnitude of A + B. 2 2 A + B is . The scalar product—Figures 1.25–1.26 • The scalar product (also called the dot product ) of two vectors r r is AgB = AB cosφ. • Figures 1.25 and 1.26 illustrate the scalar product. Copyright © 2012 Pearson Education Inc. Q1.10 Consider the two vectors ! A = 3iˆ + 4 ˆj ! B = −8iˆ + 6 ˆj ! ! What is the dot product A • B ? A. zero B. 14 C. 48 D. 50 E. none of these © 2012 Pearson Education, Inc.
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