WK 2 Homework – due Friday, 9/16
Reading assignment:
•
1.7 – 1.9
•
Posted notes on website
•
Reading question: 1.13; 1.16
Questions: 1.32, 31, 38, 41, 55, 59, 68 – the solutions are
on the school website.
Homework – due Tuesday, 9/20 – 11:00 pm
Mastering physics wk 2
Vectors
• A vector is a mathematical entity that possesses two
properties, which physically we call magnitude and direction.
displacement, velocity, acceleration, force, and momentum are
vector quantities.
• A scalar is a mathematical entity that has one property,
magnitude, only. Temperature, mass, speed, and energy are
scalar quantities. Scalars obey the familiar rules of addition,
multiplication, etc.
• The operation of vector include:
– Addition,
– Subtraction,
– Multiplication
• by a scalar,
• by a vector:
– Dot product,
– Cross product
• Two ways to represent vectors
– Geometric approach
• Vectors are symbolized graphically as arrows, in
text by bold-face type or with a line/arrow on top.
A
y
y1
o
– Algebraic approach
• To describe motion of an ideal particle, we choose
a coordinate system, e.g. Cartesian x, y, z. The
system must be an inertial coordinate system,
which means it is non-accelerated.
p(x1, y1)
x1
x
Representing vector geometrically
• If two vectors have the same size and same
direction, they are equal.
=
=
=
• If two vectors have the same size but opposite
direction, then we say one vector is the negative
of the other.
A
B
A
=-
B
Adding vectors
  
A B  C
• The sum of two vectors is also a vector.
• Graphical method: Vectors are represented by arrows,
drawn to scale. Place the tail of the 2nd vector on the
head of the 1st, preserving the relative orientations. The
resultant vector extends from the tail of the 1st to the
head of the 2nd vector.
• The sum of two vectors is also a
vector.
• Addition makes sense only for
same kinds of vectors
The order of addition does not matter!
parallelogram
Head and tail method
Parallelogram method
Adding 3 or more vectors
Subtract vectors: adding a
negative vector
Multiplication: scalar x vector
representing vector algebraically
• A vector can be completely describe by its
components in a coordinate system. The
origin of the systems is the tail of vector
r is a position vector
from the origin to the
point x, y, z
The position vector r of the point P.
Unit vectors
• A unit vector is a vector that has a magnitude of 1, with
no units. Its only purpose is to point, or describe a
direction in space.
• Unit vector is denoted by “^” symbol.
• For example:
– ^
i represents a unit vector that points in the direction
of the + x-axis
y
j unit vector points in the + y-axis
– ^
^
j
– ^
k unit vector points in the + z-axis
A unit vector in any direction is represented by:

 • where a is the magnitude of the z
a
vector a . Often, the magnitude of a
aˆ  
a
vector is indicated by the letter
without the line on top: a  a
^
k
^
i
x
• Any vector can be expressed with unit vectors:
A = Ax ^
i + Ay ^
j + Az ^
k
The components Ax, Ay, Az of an
arbitrary vector A.
In 2 dimensions
Addition/subtraction of vectors algebraically
Example: inclined plane
example
a. Is the vector A = ^
i + ^
j + ^
k a unit vector?
b. Can a unit vector have any components
with magnitude greater than unity? Can it
have any negative components?
^
^
A
=
a
(3.0
i
+
4.0
j ), where a is a
c. If
constant, determine the value of a that
makes A a unit vector.
Example - Finding components
•
What is the x- and y- components of vector D in a)
where the magnitude of the vector is D = 3.00 m and
the angle θ = 45o?
What is the x- and y- components of vector E in b)
where the magnitude of the vector is E = 4.50 m and
the angle θ = 37o?
•
y
Dx (+)
Ex (+)
x
Ey (+)
θ=45o
Dy (-)
a)
θ=37o
D
E
b)
y
x
example
•
a.
b.
c.
d.
Let the angle θ be the angle that the vector A makes with
the + x-axis, measured counter clockwise from that axis.
Find the angle θ for a vector that has the following
components:
Ax = 2.00 m; Ay = -1.00 m
Ax = 2.00 m; Ay = 1.00 m
Ax = -2.00 m; Ay = 1.00 m
Ax = -2.00 m; Ay = -1.00 m
Example – adding vectors with
components
• Three players on a reality TV show are brought to the
center of a large, flat field. Each is given a meter stick, a
compass, a calculator, a shovel, and the following three
displacements;
– 74.4 m 32.0o east of N
– 57.3 m, 36.0o south of west
– 17.8 m straight south
• The three displacement lead to the point where the keys
to a new Porsche are buried. Two players start measuring
immediately, but the winner first calculates where to go.
What does she calculate?
R=A+B+C
Example – using unit vectors
Given the two displacement
A =(6 ^
i + 3^
j -^
k) m
^
^
B =(4i - 5 j + 8 ^
k) m
• Find the magnitude of the displacement
2A - B
Multiplying vectors
• There are two kinds of products of vectors
– Scalar product or dot product, yields a
result that is a scalar quantity
– Vector product or cross product, yields
another vector
Scalar product or dot product
Proof:
example
Application of scalar product
• When a constant force F is applied to a body that
undergoes a displacement d, the work done by the force
is given by
W = F∙d
The work done by the force is
• positive if the angle between F and d is between 0 and
90o (example: lifting weight)
• Negative if the angle between F and d is between 90o
and 180o (example: stop a moving car)
• Zero and F and d are perpendicular to each other
(example: waiter holding a tray of food while walk
around)
Finding the angles with the scalar product
• Find the dot product and the angle
between the two vectors
AxBx + AyBy + AzBz
A · B = |A||B|cosθ =
|A| = √Ax2 + Ay2 + Az2
|B| = √Bx2 + By2 + Bz2
cosθ =
A·B
|A||B|
=
AxBx + AyBy + AzBz
(√Ax2 + Ay2 + Az2 )(√Bx2 + By2 + Bz2 )
•if cosθ is negative, θ is between 90o and 180o
example
?
?
example
• Find the scalar product A∙B of the two vectors in the
figure. The magnitudes of the vectors are A = 4.00 and B
= 5.00
B
y
130o
A
θ 53.0o
x
Finding the angles with the scalar product
• Find the dot product and the angle between the
two vectors
A = 2i + 3j + k
B = -4i +2j - k
Vector product or cross product
• The vector product of two vectors A and
B, also called the cross product, is
denoted by A x B.
• The vector product is a vector. It has a
magnitude and direction
The direction of a vector (cross) product
Place the vector tail to tail, they define
the plane
AxB
B
θ
A
A x B is perpendicular to the plane
containing the vectors A and B.
It direction is determined by the righthand rule.
Right-hand rule: That is, if we follow
the direction of the fingers to go from
the A to B, then the thumb points in
the direction of A x B
AxB
B
θ
BxA=-AxB
A
θ
A
B
BxA
Magnitude of a vector (cross) product
A x B = ABsinθ
Where θ is the angle from A toward B, and θ is the
smaller of the two possible angles.
Since 0 ≤ θ ≤ 180o, 0 ≤ sinθ ≤ 1, A x B is never negative.
Therefore, the magnitude of a vector product is never
negative.
Note when A and B are in the same direction or in the
opposite direction, sinθ = 0;
The vector product of two parallel or anti-parallel
vectors is always zero
Vector product vs. scalar product
• Vector product:
– A x B = ABsinθ (magnitude)
– Direction: right-hand rule-perpendicular to the A, B
plane
• Scalar product:
– A∙B = ABcosθ (magnitude)
– It has no direction.
• When A and B are parallel • When A and B are
perpendicular to each other
– AxB is zero
– A∙B is maximum
– AxB is maximum
– A∙B is zero
Calculating the vector product using
components
• If we know the components of A and B, we can
calculate the components of the vector product.
• The product of any vector with itself is zero
＊i x i = 0; j x j = 0; k x k = 0
• Using the right hand rule and A x B = ABsinθ
＊i x j = -j x i = k;
＊j x k = -k x j = i;
＊k x i = - i x k = j
A x B = (Axi + Ayj + Azk) x (Bxi + Byj + Bzk)
= AxByk - AxBzj
– AyBxk + AyBzi
+ AzBxj - AzByi
A x B = (AyBz – AzBy) i + (AzBx - AxBz) j + (AxBy – AyBx) k
If C = A x B then
Cx = AyBz – AzBy;
Cy = AzBx - AxBz;
Cz = AxBy – AyBx
The vector product can also be expressed in determinant
form as
AxB=
i
j
k
i
j
k
Ax
Ay
Az
Ax
Ay
Az
Bx
By
Bz
Bx
By
Bz
- direction
+ direction
A x B =(AyBz – AzBy) i + (AzBx - AxBz) j + (AxBy – AyBx) k
Example 1.12
• Vector A has magnitude 6 units and is in the direction of
the + x-axis. Vector B has magnitude 4 units and lies in
the xy-plane, making an angle of 30o with the + x-axis
(fig. 1.32). Find the vector product C = A x B.
y
B
30o
A
C
z
x
example
• Find the vector product A X B (expressed in unit vectors)
of the two vectors given in the figure.
A (3.60 m)
70o
30o
B (2.4 m)
Class work Homework – due Friday, 9/23
Reading assignment:
• 12.6; 2.1-2.6
Questions: 2, 7, 9, 12, 13, 22, 23, 24, 25, 29,
31, 34, 40, 44 – the solutions are on the
school website.
Homework – due Tuesday, 9/20 –
11:00 pm
Mastering physics wk 3