Transcript Vectors - Fundamentals and Operations

Vectors - Fundamentals and Operations
A vector quantity is a quantity
which is fully described by both
magnitude and direction.
Vector quantities are often represented by scaled vector diagrams.
Vector diagrams
depict a vector
by use of an
arrow drawn to
scale in a specific
direction.
There are several characteristics
of this diagram which make it an
appropriately drawn vector diagram.
A. A scale is clearly listed
B. An arrow (with arrowhead) is drawn
in a specified direction; thus, the
vector has a head and a tail.
C. The magnitude and direction of the vector is clearly labeled;
in this case, the diagram shows the magnitude is 20 m and the
direction is (30 degrees West of North).
Vector Components
A vector is a quantity which has
magnitude and direction.
Displacement, velocity,
acceleration, and force are the
vector quantities which we will
discuss in our course.
Resultants
The resultant is the vector sum of two or more
vectors. It is the result of adding vectors
together.
Conventions for Describing Directions of Vectors
Vectors can be directed due East, due West, due South, and
due North. But some vectors are directed northeast (at a 45
degree angle)
An example of the use of the
head-to-tail method is illustrated
below. The problem involves the
addition of three vectors:
20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.
SCALE: 1 cm = 5 m
The head-to-tail method is employed as
described above and the resultant is
determined (drawn in red). Its magnitude and
direction is labeled on the diagram.
SCALE: 1 cm = 5 m
Interestingly enough, the order in which three vectors
are added is insignificant; the resultant will still have
the same magnitude and direction. For example, consider
the addition of the same three vectors in a different
order.
15 m, 210 deg. + 25 m, 300 deg. + 20 m, 45 deg.
SCALE: 1 cm = 5 m
When added together in this different order, these same
three vectors still produce a resultant with the same
magnitude and direction as before (22 m, 310 deg.). The
order in which vectors are added using the head-to-tail
method is insignificant.
SCALE: 1 cm = 5 m
Example
The parallelogram method of vector resolution involves
using an accurately drawn, scaled vector diagram to
determine the components of the vector. Briefly put, the
method involves drawing the vector to scale in the indicated
direction, sketching a parallelogram around the vector such
that the vector is the diagonal of the parallelogram, and
determining the magnitude of the components (the sides of
the parallelogram) using the scale
The trigonometric method of vector
resolution involves using trigonometric
functions to determine the components of
the vector.
trigonometric functions relate
the length of the sides of a
right triangle to the angle of a
right triangle. As such,
trigonometric functions can be
used to determine the length
of the sides of a right triangle
if one angle and the length of
one side are known.
The Pythagorean theorem is
a useful method for
determining the result of
adding two (and only two)
vectors which make a right
angle to each other. The
method is not applicable for
adding more than two
vectors or for adding
vectors which are not at 90degrees to each other. The
Pythagorean theorem is a
mathematical equation which
relates the length of the
sides of a right triangle to
the length of the
hypotenuse of a right
triangle.
Oblique Triangles
An oblique triangle is any triangle that is not a right triangle.
It could be an acute triangle (all threee angles of the triangle
are less than right angles) or it could be an obtuse triangle
(one of the three angles is greater than a right angle).
Solving oblique triangles
The trigonometry of oblique triangles is not as simple of that
of right triangles, but there are two theorems of geometry
that give useful laws of trigonometry. These are called the "law
of cosines" and the "law of sines."
The law of cosines
This is a simply stated equation:
c2 = a2 + b2 – 2ab cos C
It looks like the Pythagorean theorem except for the
last term, and if C happens to be a right angle, that
last term disappears (since the cosine of 90° is 0), so
the law of cosines is actually a generalization of the
Pythagorean theorem.
Note that each triangle gives three equations for the law
of cosines since you can permute the letters as you like.
The other two versions are then
a2 = b2 + c2 – 2bc cos A, and
b2 = c2 + a2 – 2ca cos B.
The law of sines
The law of sines is also a simply stated equation
sin a/a = sin b/b = sin c/c
You can use the law of sines in two ways
First, if you know two angles and the side opposite one of
them, then you can determine the side opposite the other
one of them. For instance, if angle A = 30°, angle B = 45°,
and side a = 16, then the law of sines says
(sin 30°)/16 = (sin 45°)/b. Solving for b gives
b = 16(sin 45°)/(sin 30°) = 22.6274.
The Plane and The Wind
Problems
http://www.glenbrook.k12.il.us/gbssci/phys/Class/vectors/u3l1a.html
http://aleph0.clarku.edu/~djoyce/java/trig/oblique.html