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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

PHY 1151 Principles of Physics I

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: 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

- 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

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 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 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 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 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) 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 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 Direction

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

PHY 1151 Principles of Physics I

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