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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
Adding Vectors Graphically
s  a b
General procedure for adding two vectors
graphically:
• (1) On paper, sketch vector a to some
convenient scale and at the proper angle.
• (2) Sketch vector b to the same scale,
with its tail at the head of vector , again at
the proper angle.
• (3) The vector sum s is the vector that
extends from the tail of a to the head of b
.
Examples
Two important properties of vector additions
(1) Commutative law:
a b  b a
(2) Associative law:
(a  b )  c  a  (b  c )
Subtraction
d  a  b  a  (b )
Check Your Understanding
Two vectors, A and B, are added by means of
vector addition to give a resultant vector R:
R=A+B. 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?
Unit Vectors
The unit vectors are dimensionless vectors that
point in the direction along a coordinate axis
that is chosen to be positive
How to describe a two-dimension vector?
Vector Components:The projection of a vector on an axis
is called its component .
ax  a cos 
a y  a sin 
a  ax  ay  ax i  ay j
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
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.
Reconstructing a Vector from
Components
;
Magnitude:
Direction:
a  ax2  a y2
 ay 
  tan  
 ax 
1
Addition of Vectors by Means of
Components
C  A B

=
  tan 1 (Cy / Cx )
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?
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.
Multiplying and Dividing a Vector by a Scalar
eV  e(Vx  Vy )  e(Vx i  Vy j )  (eVx )i  (eVy ) j
The Scalar Product of Vectors
(dot product )
•The dot product is a scalar.
•If the angle between two
vectors is 0°, dot product is
maximum
•If the angle between two
vectors is 90°, dot product is
zero
The commutative law
Example
What is the angle between
and
?
The Vector Product (cross product )
c  a b
(1) Cross production
is a vector
(2) Magnitude is
c  ab sin 
(3) Direction is
determined by
right-hand rule
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.
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
?
Sample Problem
If
what is
and
?
,