Prof.
Clements Notes/Hints for Physics Courses
Chapter
3 Open Stax College Physics or Most Introductory Physics Courses
Vectors,
Vector Addition, Subtraction, Multiplication
I
highly recommend that you go to the web site
https://openstax.org/details/books/college-physics
to access the free OpenStax College Physics textbook and resources.
YouTube
videos of my (usually short) lectures and example problems are
indexed at http://www.physics.gpclements.com/
. There are also a few videos that give review of basic math tools
that are used in an introductory physics course.
Terms
you should know at the end of this unit: vector, vector
addition and subtraction, negative vector, scalar multiplication
This
chapter will move you toward understanding and solving problems where
objects move in two dimensions. This chapter will also help your
understanding of vectors and vector related calculations. This
chapter is not a major step up in difficulty compared to the previous
chapter. You will still use the kinematic equations and your
understanding of position, velocity, and acceleration. You will need
to understand how to work with right triangles and trigonometric
functions. Be sure to get help (your instructor, class mates,
YouTube videos) if you have lingering questions after you study this
material.
Introduction
In
the early chapters of physics textbooks you will usually only study
problems in which the motion is over a small area and you can
consider the Earth to be flat. I will not discuss problems in three
dimensions.
The
following is a very important concept. You must NEVER directly add
numbers that are not in the same (straight line) direction. i.e. For
displacement, you must NEVER add 4 km in the North direction to 3 km
in the East direction to produce a result of 7 km. This result is
wrong.
I
will talk about X-type numbers and Y-type numbers. We will have two
sets of equations in our problems. One set of equations will apply
to motion in the X direction. A separate set of equations will apply
in the Y direction. Another way to state this concept is that motion
in the X direction has no effect on motion in the Y direction. i.e.
An acceleration in the X direction has no effect on velocity in the Y
direction. This point will be covered again later in the chapter.
It
is common for textbooks to show illustrations that verify that motion
in the X direction has no effect on motion in the Y direction.
Graphical
Vector Addition and Subtraction
You
should review the description of vectors in your textbook. A vector
quantity has some magnitude and a direction. e.g. A value of
velocity might be 4 m/s at an angle of 30 degrees to the positive X
axis. Textbooks often show vector symbols in boldface or with an
arrow above the symbol of the quantity.
The
two methods of adding vectors are 1) graphical and 2) analytical
(mathematical). This post will discuss the graphical method. The
following post will discuss the analytical addition of vectors.
The
tail of a vector is the starting part of the arrow. The head of the
vector is the arrow part. For the graphical method you:
1)
draw the first vector with an approximately correct length and the
approximately correct angle to the X axis.
2)
draw the second vector where the first vector ended. This creates
the name of the method: "head-to-tail." Draw the second
vector with its approximately correct angle relative to a line that
is parallel to the X axis.
3)
draw the resultant vector as a straight line starting at the
beginning (tail) of the first vector that was drawn and ending at the
head (arrow) of the last vector that was drawn.
You
should review the examples of vector addition in your textbook. The
graphical method is always approximate.
A
minus sign in front of a vector symbol indicates that the direction
of the vector is changed by 180 degrees while the magnitude is
unchanged.
Algebra
allows performing subtraction through the addition process. Suppose
we want to subtract 3 from 8. We could write 8 - 3 or 8 +
(-3) and produce the result of 5 either way. Vector subtraction is
done by adding a negative vector. The method is similar to the
calculation 8 + (-3) . Review the examples of vector subtraction in
your textbook. The image below shows the addition and subtraction of
two vectors:
C
= A + B C =
A + (- B )
Multiplication
of a vector by a scalar:
What
is the meaning of 3 * 5? Answer: 3 * 5 is 5 + 5 + 5.
What
is the meaning of 3X? Answer: 3X is X + X + X.
What
is the meaning of 3A ? Answer: 3A is A +
A + A .
Multiplication
of a vector by a scalar just makes the vector longer. If the scalar
is positive the direction is unchanged. The image above shows the
vector multiplication 2A. If the scalar is negative the
vector switches direction by 180 degrees.
Copyright©
2017 by Greg Clements Permission is granted to reproduce this
document as long as 1) this copyright notice is included, 2) no
charge of any kind is made, and, 3) the use is for an educational
purpose. Editing of the document to suit your own class style and
purposes is allowed.
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