chapter 3
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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 - PowerPoint PPT PresentationTRANSCRIPT
PHY 1151 Principles of Physics I
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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
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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.
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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.
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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.
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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.
x
y
III
III IV
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Example: Find the Components of a Vector 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°.
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Find the Magnitude and Direction of A Given its Components Ax and Ay
The magnitude and direction of A are related to its components through the expressions: A = (Ax
2 + Ay2)1/2
= tan-1(Ay/Ax) Note: Pay attention to the
signs of Ax and Ay to find the correct values for .
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Example: Find the Magnitude and Direction of a Vector 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.
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Express Vectors Using Unit Vectors
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
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Adding Vectors Using the Components Method 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.
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Example: The Sum of Two Vectors (with Components Method)
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.
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Example: Adding Vectors Using Components
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.
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