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Guidelines For Vector Addition And Subtraction
A vector is a geometrical figure that represents a vector quantity. A vector quantity is one that has
magnitude (size) and direction (e.g. displacement or velocity). The geometric figure that is a
vector is nothing more than an arrow. The length of the arrow represents the magnitude of the
vector quantity, while the direction of the arrow from head to tail represents the direction of the
vector quantity.
This is a vector that may represent a displacement of 5 km east.
tail
head
ADDING VECTORS
1. Graphical Method
A. Scale selection
1. Select a starting point. (maybe the origin of an x-y coordinate system.)
2. Select a directional scale. Often the directional scale is given such as North, South, East and
West. In this case, North is usually pointed toward the top of your paper, South - bottom,
etc.
In this case, directions (angles) are often given as North, South, East and West. from a
starting point. Look at the examples below.
N
V = 5 km @ 30 ° North of West
V
30 °
E
Note: Only if a vector falls
exactly between North and
East (45 °) can we call it
North East.
If no directional scale is given, only degrees, then normal Cartesian coordinates are used
with 0 ° being the positive x - axis and 90° being the positive y - axis.
V
V = 5 km @ 150 °
30 °
4. Select a magnitude scale which will be convenient to the magnitudes of the vectors given.
for example: 1 mm = 2 meters/sec or 1 cm = 100 Newtons.
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Guidelines For Vector Addition And Subtraction
B. Adding the vectors: Tail to Head Method
1. Place the tail of the first vector (arrow) at the starting point (origin). Use a protractor to
carefully measure the direction. Draw a vector in the direction given with the length equal
to the magnitude scale chosen. Place an arrow head at the head of the vector. (this is a
vector after all).
2. At the head of the first vector, draw a mini coordinate axis (exactly as the original one) so
you can find the direction of the next vector.
3. Starting at the head of the first vector, draw in the second vector using the new coordinate
axis for direction and the magnitude scale for length.
4. Continue drawing additional vectors (if there are more given) by placing the tail of the next
vector on the head of the previous vector, until all vector are drawn.
5. Determine the resultant (sum) of the vectors, by drawing another vector from the origin
point (tail of the first) to the head of the last vector drawn. The magnitude of the resultant
is the scaled length of this vector, while the direction can be measured from the original
coordinate axis.
e.g.: add the following displacements: V1 = 4 km @ 45 ° and V2 = 6 km @ 270 °
so V1 + V2 = R
Notes:
a) Vectors obey the commutative property of addition.
b) A negative Vector is one with the same magnitude but opposite direction (180° difference). So the
easiest way to subtract vectors, is to add the negative.
c) An alternate method, called the Parallelogram Method can be learned and used, but is only
effective when only two vectors are being added. Graphically, this method can be extremely
clumsy.
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