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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

Chapter 3

Vectors in Physics (Continued)

PHY 1151 Principles of Physics I


Outline
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

  • Adding vectors using the Components Method

PHY 1151 Principles of Physics I


Components method for adding vectors
Components Method for Adding Vectors

  • 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
Components of a Vector

  • 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


Find the components of a vector given its magnitude and direction
Find the Components of a Vector Given its Magnitude and Direction

  • 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


Signs of the components a x and a y

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


Example: Find the Components of a Vector Direction

  • 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 a given its components a x and a y
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


Example find the magnitude and direction of a vector
Example: Find the Magnitude and Direction of a Vector Direction

  • 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
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


Adding vectors using the components method
Adding Vectors Using the Components Method Direction

  • 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


Example the sum of two vectors with components method
Example: The Sum of Two Vectors (with Components Method) Direction

  • 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


Example adding vectors using components
Example: Adding Vectors Using Components Direction

  • 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
Homework Direction

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

PHY 1151 Principles of Physics I


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