Chapter 3

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Chapter 3

Vectors in Physics (Continued)

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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.

o

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Homework Chapter 3, Page 73, Problems: #4,

8, 14, 21, 26.