Vectors
Vector quantities are often represented
by scaled vector diagrams.
Vector diagrams must include:
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a scale
an arrow drawn in a specified
direction; thus, the vector has a
head and a tail.
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, or N30W).
There are Two Ways of Identifying the Direction of a Vector:
Observe in the second example that the vector is said to have a direction of 240
degrees. This means that the tail of the vector was pinned down, and the vector
was rotated an angle of 240 degrees in the counterclockwise direction beginning
from due east
Scale Vector Diagrams
The magnitude of a vector in a scaled vector diagram is depicted by the length of
the arrow. The arrow is drawn a precise length in accordance with a chosen scale.
For example, a vector with a magnitude of 20 miles using the scale 1 cm = 5
miles, has a vector arrow with a length of 4 cm. That is, 4 cm x (5 miles/1 cm) =
20 miles.
Resultants
The resultant is the vector sum of two or more vectors.
It is the result of adding vectors together. If displacement vectors A, B, and C are
added together, the result will be vector R.
We say that vector R is the resultant displacement of displacement vectors A, B,
C.
Vector Addition
Observe the following summations of two vectors:
Note: you can have a negative vector!
Think of these like integers, if you travel
backwards the way you came and pass your
origin you can have a negative
displacement. Likewise you can have
negative magnitudes for distance due to
their direction.
The forwards direction (the first direction)
is positive and the backwards direction is
always negative.
If you are not given this information assume that north and east are the
positive values and south and west are the negative ones.
There are a variety of methods for determining the magnitude and direction of the
result of adding two or more vectors.
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the Pythagorean theorem and trigonometric methods
the head-to-tail method using a scaled vector diagram
Head-to-Tail Method
The following step-by-step method for applying the head-to-tail method to
determine the sum of two or more vectors is given below.
1. Choose a scale (the best choice of scale is one which will result in a diagram
which is as large as possible, yet fits on the sheet of paper).
2. Pick a starting location and draw the first vector to scale in the indicated
direction. Label the magnitude and direction of the scale on the diagram
(e.g., SCALE: 1 cm = 20 m).
3. Starting from where the head of the first vector ends, draw the first vector to
scale in the indicated direction. Label the magnitude and direction of the
vector on the diagram.
4. Repeat steps 2 and 3 for all vectors which are to be added
5. Draw the resultant from the tail of the first vector to the head of the last
vector. Label this vector as "Resultant" or simply "R."
6. Using a ruler, measure the length of the resultant and determine its
magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm =
88 m).
7. Measure the direction of the resultant.
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
SCALE: 1 cm = 5 m
The order in which three vectors are added is insignificant; the resultant will still
have the same magnitude and direction.
Example:15 m, 210 deg. + 25 m, 300 deg. + 20 m, 45 deg.
SCALE: 1 cm = 5m
These same three vectors still produce a
resultant with the same magnitude and
direction as before (22 m, 310 deg.).
SCALE: 1 cm = 5 m
Method 2: Pythagorean Theorem
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. It will only
work for right-angled triangles.
To see how the method works, consider the following problem:
A hiker leaves camp and hikes 11 km, north and then hikes 11 km east. Determine
the resulting displacement of the hiker.
Example:
Answer A:
Answer B:
R2 = (5)2 + (10)2
R2 = (30)2 + (40)2
R2 = 125
R2 = 2500
R = SQRT (125)
R = SQRT (2500)
R = 11.2 km
R = 50 km
Method 3: Basic Trigonometry
Note: this method is considered advanced, but it ultimately makes problem
solving easier.
Trigonometry assumes that all right angled triangles have a common ratio of sides
vs. interior angles.
The three equations below summarize these three functions in equation form.
SIN = O/H
COS = A/H
TAN = O/A
REMEMBER SOH CAH TOA
THERE IS A SIN, COS AND TAN BUTTON ON YOUR CALCULATOR.
PRESS SIN THEN THE ANGLE TO FIND WHAT B/C EQUALS (IT WILL
BE A DECIMAL) PRESS SHIFT SIGN OR SIN INVERSE TO GO FROM
A DECIMAL TO THE ANGLE.
REMEMBER THIS ONLY WORKS FOR RIGHT ANGLED TRIANGLES!
Try this on your calculator to make sure you get the right angle. Then go
backwards and make sure you get 45 degrees.