Chapter 3. Vector

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Chapter 3. Vector

• 1. Adding Vectors Geometrically      
• 2. Components of Vectors      
• 3. Unit Vectors      
• 4. Adding Vectors by Components        
• 5. Multiplying Vectors 

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Adding Vectors Graphically

• General procedure for adding two vectors
  graphically
• (1) On paper, sketch vector to some
  convenient scale and at the proper angle.
• (2) Sketch vector to the same scale, with
  its tail at the head of vector , again at the
  proper angle.
• (3) The vector sum is the vector that extends
  from the tail of to the head of .

• General procedure for adding two vectors
  graphically
• (1) On paper, sketch vector to some
  convenient scale and at the proper angle.
• (2) Sketch vector to the same scale, with
  its tail at the head of vector , again at the
  proper angle.
• (3) The vector sum is the vector that extends
  from the tail of to the head of .

• General procedure for adding two vectors
  graphically
• (1) On paper, sketch vector to some
  convenient scale and at the proper angle.
• (2) Sketch vector to the same scale, with
  its tail at the head of vector , again at the
  proper angle.
• (3) The vector sum is the vector that extends
  from the tail of to the head of .

• General procedure for adding two vectors
  graphically
• (1) On paper, sketch vector to some
  convenient scale and at the proper angle.
• (2) Sketch vector to the same scale, with
  its tail at the head of vector , again at the
  proper angle.
• (3) The vector sum is the vector that extends
  from the tail of to the head of .

• General procedure for adding two vectors
  graphically
• (1) On paper, sketch vector to some
  convenient scale and at the proper angle.
• (2) Sketch vector to the same scale, with
  its tail at the head of vector , again at the
  proper angle.
• (3) The vector sum is the vector that extends
  from the tail of to the head of .

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Examples
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Two important properties of vector additions
(1) Commutative law
(2) Associative law
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Subtraction
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Check Your Understanding 

• Two vectors, A and B, are added by means of
  vector addition to give a resultant vector R
  RAB. The magnitudes of A and B are 3 and 8 m,
  but they can have any orientation. What is
• (a) the maximum possible value for the
  magnitude of R?
• (b) the minimum possible value for the
  magnitude of R?

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Unit Vectors

• The unit vectors are dimensionless vectors that
  point in the direction along a coordinate axis
  that is chosen to be positive

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How to describe a two-dimension vector?

• Vector ComponentsThe projection of a vector on
  an axis is called its component .

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Properties of vector component

• The vector components of the vector depend on the
  orientation of the axes used as a reference.
• A scalar is a mathematical quantity whose value
  does not depend on the orientation of a
  coordinate system. The magnitude of a vector is a
  true scalar since it does not change when the
  coordinate axis is rotated. However, the
  components of vector (Ax, Ay) and (Ax', Ay'),
  are not scalars.
• It is possible for one of the components of a
  vector to be zero. This does not mean that the
  vector itself is zero, however. For a vector to
  be zero, every vector component must individually
  be zero.
• Two vectors are equal if, and only if, they have
  the same magnitude and direction

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Example 1  Finding the Components of a Vector
A displacement vector r has a magnitude of r 175
m and points at an angle of 50.0 relative to the
x axis in Figure. Find the x and y components of
this vector.
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Reconstructing a Vector from Components

Magnitude
Direction
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Addition of Vectors by Means of Components
q

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Check Your Understanding 

• Two vectors, A and B, have vector components that
  are shown (to the same scale) in the first row of
  drawings. Which vector R in the second row of
  drawings is the vector sum of A and B?

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Example 2  The Component Method of Vector Addition

• A jogger runs 145 m in a direction 20.0 east
  of north (displacement vector A) and then 105 m
  in a direction 35.0 south of east (displacement
  vector B). Determine the magnitude and direction
  of the resultant vector C for these two
  displacements.

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Multiplying and Dividing a Vector by a Scalar
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The Scalar Product of Vectors(dot product )
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The commutative law
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Example

• What is the angle between
• and ?

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The Vector Product (cross product )
(3) Direction is determined by right-hand rule
(1) Cross production is a vector
(2) Magnitude is
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Property of vector cross product

• The order of the vector multiplication is
  important.

If two vectors are parallel or anti-parallel,
             .
If two vectors are perpendicular to each other ,
the magnitude of their cross product is maximum.
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Sample Problem
In Fig. 3-22, vector     lies in the xy plane,
has a magnitude of 18 units and points in a
direction 250 from the x direction. Also,
vector     has a magnitude of 12 units and
points in the z direction. What is the vector
product                 ?
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Sample Problem
If                      and                ,
what is                         ?