Vectors
Some quantities can be described with only a number. These
quantities have magnitude (amount) only and are referred to as
scalar quantities.
Scalar - a quantity which can be described with only a
magnitude
Quantities that are described with both a magnitude and a
direction are called vector quantities.
Vector - a quantity with both magnitude and direction
Vectors
We can use a graphical method to help us picture vectors, and
to allow us to understand how vectors are added and
subtracted. We will represent vector quantities with arrows.
The length of the arrow represents its magnitude, the arrow tip
and the direction of the arrow represent the vector direction.
tail
tip
The point of the arrow is referred to as the “Tip”. The nonpointed end is called the “Tail”
Vectors
The length and direction gives all the info about a vector.
Vectors A and B are completely equivalent.
B = 12 m east
A = 12 m east
A=B
A vector can be moved (translated) without changing it. As
long as the length and direction do not change, the vector has
not changed.
Vectors
If vectors are to be worked with graphically, axes and an
appropriate scale must be chosen. Suppose we are describing
the motion of a bug walking on the ground. If the bug walks 12
m to the east, we call this the bug’s displacement. We would
write this as A = 12 m east . Once we establish a coordinate
system, we can represent this displacement vector with an
N
arrow like the one below.
W
E
A = 12 m east
S
Vector Addition
The process of adding two or more vectors is called Vector
Addition.
The sum of two or more vectors is called the resultant vector.
Vector Addition
“Triangle Method”
Steps for adding two vectors
using the tip-to-tail method:
of Vector Addition
N
1) Draw the first vector (A) to
scale, with its tail at the origin
C
B
W
A
S
C=A+B
E
2) Draw the second vector (B)
also to scale, with its tail at the
tip of the first vector
3) The resultant vector runs
from the tail end of the first
vector to the tip of the second
vector.
Vector Addition
“Parallelogram Method”
Steps for adding two vectors using the
parallelogram method:
of Vector Addition
N
C
W
B
E
A
S
C=A+B
1) Draw the first vector (A) to scale,
with its tail at the origin
2) Draw the second vector (B) also
to scale, and also with its tail at the
origin.
3) Starting at the tip of one vector,
draw a dotted line parallel to the
other vector. Repeat, starting from
the tip of the second vector.
4) The resultant vector runs from the
origin to the intersection of the two
dotted lines.
Vector Addition
Notice - whichever method you use, you get the same value for
vector C !
N
N
C
C
B
W
E
W
B
A
A
S
S
C=A+B
C=A+B
Tip - to - Tail Method
E
Parallelogram Method
Vector Addition
The methods work no matter the direction of the original vectors.
Tip - to - Tail Method
Parallelogram Method
N
N
B
W
B
C
A
E
W
C
A
S
S
A+B=C
A+B=C
E
Vector Addition
If you need to add more than two vectors, the tip-to-tail method is
usually easiest.
Note that the order the vectors
are added does not affect the
result.
N
N
C
A
D
B
B
W
E
W
D
C
E
A
S
A+B+C=D
S
C +B +A=D
Vector Subtraction
For any vector A, the vector -A is simply a vector equal in
magnitude and opposite in direction to A.
A
-A
Vector Subtraction
Subtracting one vector from another is the same as adding a
negative vector. A - B = A + (-B)
addition: A + B
A
B
A+B=C
B
C
A
subtraction: A - B
A
-B
A + (-B) = D
-B
A
D
Vector addition: Mathematical Method
To add two perpendicular vectors
mathematically, make a sketch of the two
vectors you are adding, and use either
graphical method to sketch their resultant.
N
C

W
A
B
E
To find the magnitude of the resultant, use
the Pythagorean theorem.
A +B =C
2
S
2
To find the direction of the resultant, use an
inverse trig function, for example: tan-1
C=A+B
Tip - to - Tail Method
2
Tan = B/A
= tan-1(B/A)
Finding Vector Components: Mathematical Method
To break a vector down into perpendicular
components, first make a sketch of the
vector.
N
C
W
CN

CE
S
C = CE + CN
E
Sketch dotted lines from the tip of the
vector and parallel to each axis.
Draw in the components, from the origin to
the dotted line.
To calculate the magnitude of the
components, use the sin and cos functions.
Sin = CN/C
CN = C sin
Cos = CE/C
CE = C cos