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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.
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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.
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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
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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.
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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.
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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
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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
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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
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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.
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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
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