Vector Notes – Dynamics
A. What are Vectors?
So far in our study of mechanics we have looked at ____________, the study of
motion without regard to how that motion is caused. We now begin our study of
dynamics or the __________ that cause motion. Representing force requires a
thorough understanding of ____________. You know that a vector has both a
_______________ and _____________. On paper, any vector can be represented by
arrows in which the __________ of the arrow represents the magnitude of the vector.
The ____________ the arrow points is the direction of the vector. Vectors are a
powerful tool that allow us to solve sophisticated problems by symbolizing and
representing various vector quantities such as f________, p___________,
d_____________, v_________, a____________ and many more.
Vector examples:
Vectors can be extremely useful to solve complex dynamics problems where a number of
forces act on a single object. Vectors allow use to calculate how these forces accelerate
objects and ____________ motion.
B. Vector Notation
Vectors are really line segments with arrows at the end for direction. Often, vectors
will include the point at the tail and head, say A and B with an arrow on top to indicate
its property as a vector. A vector can also be given a single letter name such as vector C
without an arrow on top.
B
C
A
Vector AB
Vector C
tail
head
C. Representing Vectors
Every vector has two parts, namely a _______ and a _______. The head of the vector
is the __________ while the tail is the ____________ point of the vector.
Vectors can be simple ___________ or they can be drawn to a specified scale. The
default direction orientation is:
Top of the page: North
N
E
W
S
For example, if a displacement of 3 km [E] is to be represented, an appropriate scale
could be used like 1 cm = 1 km. The actual vector would be _____ cm long in the desired
direction.
(actual length 3 cm)
(Actual length 5 cm)
5 km [W] could be represented as:
Directions are expressed in _____________ brackets. When a vector is exactly
halfway between (_____º to either direction) a cardinal direction, it is given the
following names: [NE] northeast, [SE] southeast, [SW] southwest, [NW] northwest
N
NE
NW
E
W
SW
SE
S
When a vector is not orientated conveniently on a NESW direction or exactly halfway in
between, we need a method by which we can properly name the vector and have someone
else replicate the identical vector with the identical angle. There are a few standard
ways to identify direction for a vector. Here is a template for the style we will use the
most in physics:
vector magnitude and unit [primary dir., ºfrom primary dir., secondary dir.]
The primary direction is the major direction the vector is closest to. For
example,
the vector below could be described as ______[__________]. In words we would say
that it is 30 m/s ____º _________ of _________. Alternately, you could describe
this vector as ______[__________]. Both are correct. Most of the time you will see
the vector described in such a way that yields an angle that is less than 45º in its
description.
N
30º
W
E
S
Write the name of each vector below on the line provided:
a.
b.
N
W
38º
15.0 N
21.0 km
22º
____________________
____________________
c.
d.
17.0 m/s2
67º
E
W
97º
36.0 m/s
D. Vector Relationships
1. When two vectors have the same magnitude and the same direction they
are considered ____________.
2. When two vectors have the same magnitude but are opposite in direction,
they are considered ______________.
Opposite vectors:
Vector w and vector x are opposite
therefore w = - x and x = - w.
w
x
E. Vector Operations
1. Adding Vectors - Like scalars, vectors can be added. When scalars are
added together, the result is called a ______. When vectors are added
together however the result is called the __________________. When adding vectors
you always connect the head of one vector to the tail of the other. The resultant will be
the vector drawn from the tail of the _______ vector to the head of the __________
vector. Here are some examples of vector addition that are sketched. ____________
vectors are vectors that meet at 90º angles.
a. collinear vectors (same direction)
+
=
=
b. collinear vectors (opposite direction)
+
=
=
c. orthogonal vectors
+
=
=
d. non-orthogonal vectors
+
=
=
Any number of vectors can be added using this method. Vectors can also be
added by using scale diagrams as already mentioned. This method can solve vector
problems _________________.Vectors can also be added
_________________. This is the method we will spend the most time on.
2. Subtracting Vectors – Like scalars, vectors can be subtracted. The
method for subtracting vectors is you always add the _____________.
a. collinear vectors (same direction)
=
+
=
b. collinear vectors (opposite direction)
=
+
=
c. orthogonal vectors
-
=
+
=
d. non-orthogonal vector
-
=
+
=
3. Adding Orthogonal Vectors:
To illustrate how vectors can solve problems let’s use an orienteering
example. Jack runs 3.0 m east then 4.0 m north. What is his overall
displacement? Let us first use the graphical method to solve this
problem then use the mathematical method. Our answers should agree
within a small degree of tolerance. Let the scale be 1 cm = 0.5 m and make
sure to use a protractor and ruler.
Graphical Method:
Mathematically: Just sketch both vectors and label them with their
appropriate magnitudes. Draw the resultant vector. Use the Pythagorean
Theorem to arrive at the magnitude of the resultant. Use trigonometry
to come up with the appropriate angle for the direction of the vector.
Resultant __________
How closely do your answers match?
4. Adding/Subtracting Non-orthogonal Vectors
Jerry walks 14.5 km [W34ºN] and then 18.7 km [E9ºS]. How far is he from his
original starting point and in what direction?
Graphical Method: Use any scale you like.
D. Resolving Vectors
We know now that vectors can be added together to produce a __________ where two
vectors become _______. The process by which one vector can be split up into two
vectors is called _______________ vectors. You will see how useful this can be for
solving dynamic problems later. The process for this is to consider the vector to be the
hypotenuse of a yet unresolved __________triangle. The two vectors that you draw
will coincide with the normal default coordinate system you are using. We will not need
to resolve vectors that have directions of due NESW already. For example, let’s say
that we want to resolve vector A below.
A
If North is the _____ of the page this vector is pointing ____________. It makes
sense then that the two vectors that would add to this resultant would be one vector
pointing _________ and the other pointing _______.
So draw them on the paper like as below. Start at the tail of the vector to draw the
first one then the head of the first vector becomes the head of the next one.
South
vector
East
Vector
Vector A’s magnitude and direction is 25.0 N [E15ºS]. Label all angles. With this
information, how can we find the magnitudes of both the south and east vectors that
add to form vector A? (Hint: use SOHCAHTOA!)
Call the horizontal part Ax. Call the vertical part Ay.
These are the _______________ of A.
Resolving Vectors Practice
1. Resolve the following vectors graphically.
2. Resolve the following vectors mathematically. Top of the page is North.
a.
b.
18º
17.0 km
36.0 m
67º
North:
South:
East:
West:
c. 35.3 km 17º W of N
d. 21.8 m 44º S of E
West:
South:
North:
East:
The Component Method
For adding non-orthogonal vectors if you don’t use the sine and cosine laws……
Steps:
2. Add the following non-orthogonal vectors using the component method.
a) 24.0 N 38º S of W (Vector A) added to 42.0 N 15º E of N (Vector B)
b) 563 km 5.0º E of S (Vector A) added to 399 km 25º E of N (Vector B)