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# Chapter 3 - PowerPoint PPT Presentation

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

Vectors in Physics (Continued)

PHY 1151 Principles of Physics I

• 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

• 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

• 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

• 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

y Direction

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

• 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

Find the Magnitude and Direction of DirectionA 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

• 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

Express Vectors Using Unit Vectors Direction

• 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

• 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

• 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

• 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

Homework Direction

• Chapter 3, Page 73, Problems: #4, 8, 14, 21, 26.

PHY 1151 Principles of Physics I