Chapter 3: Vectors
Things to ask the class:
Have you all checked your WebCT account?
Do the Quiz, participate in the discussion and learn from
the applets!
•
In this chapter we will learn about vectors, (properties,
addition, components of vectors and multiplying)
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Vectors: Magnitude and direction
Scalars: Only Magnitude
A scalar quantity has a single value with an appropriate unit and has no direction.
Examples for each:
Vectors:
Scalars:
Motion of a particle from A to B along an
arbitrary path (dotted line). Displacement is a
vector
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Vectors:
• Represented by arrows (example displacement).
• Tip points away from the starting point.
• Length of the arrow represents the magnitude
• In text: a vector is often represented
in bold face (A)

or by an arrow over the letter. A

• In text: Magnitude is written as A or A
This four vectors are equal because they
have the same magnitude and same length
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Adding vectors:
Graphical method (triangle method):
Draw vector A.
Draw vector B starting at the tip of vector A.
The resultant vector R = A + B is drawn from the tail
of A to the tip of B.
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Adding several vectors together.
Resultant vector
R=A+B+C+D
is drawn from the tail of
the first vector to the tip of
the last vector.
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Commutative Law of vector addition
A+B=B+A
(Parallelogram rule of addition)
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Dr. Mohamed S. Kariapper - KFUPM
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Associative Law of vector addition
A+(B+C) = (A+B)+C
The order in which vectors are added together does not
matter.
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Dr. Mohamed S. Kariapper - KFUPM
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Negative of a vector.
The vectors A and –A have the same magnitude but
opposite directions.
A + (-A) = 0
A
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-A
Dr. Mohamed S. Kariapper - KFUPM
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Subtracting vectors:
A - B = A + (-B)
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Dr. Mohamed S. Kariapper - KFUPM
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Multiplying a vector by a scalar
The product mA is a vector that has the same direction
as A and magnitude mA.
The product –mA is a vector that has the opposite
direction of A and magnitude mA.
Examples: 5A;
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-1/3A
Dr. Mohamed S. Kariapper - KFUPM
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Components of a vector
The x- and y-components of a
vector:
Ax  A cos 
Ay  A sin 
The magnitude of a vector:
A
Ax  Ay
2
2
The angle  between vector and x-axis:
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 Ay 
  tan  
 Ax 
Dr. Mohamed S. Kariapper - KFUPM
1
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The signs of the components Ax and Ay depend on
the angle  and they can be positive or negative.
(Examples)
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Unit vectors
• A unit vector is a dimensionless vector having a magnitude 1.
• Unit vectors are used to indicate a direction.
• i, j, k represent unit vectors along the x-, y- and z- direction
• i, j, k form a right-handed coordinate system
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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The unit vector notation for
the vector A is:
A  Axiˆ  Ay ˆj
OR in even better shorthand
notation:
A  Ax , Ay
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Adding Vectors by
Components
We want to calculate:
R=A+B
From diagram:
R = (Axi + Ayj) + (Bxi + Byj)
R = (Ax + Bx)i + (Ay + By)j
Rx = Ax + Bx
Dr. Mohamed
RyS. =Kariapper
Ay +- KFUPM
By
The components of R:
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Adding Vectors by
Components
The magnitude of a R:
R  Rx  Ry  ( Ax  Bx ) 2  ( Ay  B y ) 2
2
2
 Ry   Ay  By 

The angle  between vector R and x-axis: tan      
 Rx   Ax  Bx 
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Checkpoint 3 (page 45)
What are the signs of x components of d1 and d2?
What are the sign of the y components of d1 and d2?
What are the signs of the x and y components of d1+d2
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Sample Problem 3-4
a  ( 4.2m  iˆ  (1.5m  ˆj
b  ( 1.6m  iˆ  ( 2.9m  ˆj
c  ( 3.7m  ˆj
What is their vector sum r which is also shown ?
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Vectors and the Laws of Physics:
This section will be left as a reading
assignment for the student.
(a) The vectors a and its components
(b) The same vector, with the axes of the coordinate
system rotated through an angle 
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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Multiplying a vector by another vector:
Two types:
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Dr. Mohamed S. Kariapper - KFUPM
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Performance Objective
1. Distinguish between vector and scalar quantities.
2. Add and subtract vectors graphically.
3. Resolve vectors into their perpendicular x and y components.
4. Add two vectors perpendicular to each other mathematically.
5. Express vectors in terms of unit vectors and their scalar components.
6. Add or subtract vectors using components.
7. Explain why vectors are used to express the laws of physics.
8. Add three or more vectors graphically.
9. Multiply a vector by a scalar.
10. Find the scalar product of two vectors.
11. Find the vector product of two vectors.
12. Solve word problems involving vectors.
26-Feb-07
Dr. Mohamed S. Kariapper - KFUPM
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