Vector Addition
Mr. Davis
Vectors can be added in one dimension or two dimensions. (technically, an
infinite number of dimensions. But we will consider only two.)
One dimensional addition of vectors is the same thing, in concept, as adding
or subtracting numbers on a number line. On a number line, we move positive
or negative. With one dimensional vectors, we may move up & down, right &
left, East & West, North & South, or with two or more vectors at the same
angle.
A
B
- (B – A)
B-A
In this case vectors A and B are pointing in opposite direction. Their sum is
B - A and its magnitude (size) can also be written as |B – A|. If B is going East
and A is going West, the resultant (what you get when you add them together)
is shown as B – A. The equilibrant is the vector that is needed to add to the
resultant to get zero. In math terms, the equilibrant is the additive inverse of the
resultant. Once you find the resultant, the equilibrant is what you need to add to
get zero. The equilibrant points in the other direction from the resultant but has
the same magnitude (length).
Vector A + Vector B = Resultant
Equilibrant + Resultant = ZERO
Two dimensional addition is more complicated. There are two general
approaches, graphical and component.
The graphical methods we have examined are two: Parallelogram and tipto-tail.
In the parallelogram method, lines parallel to two given vectors are found.
Their intersection is a point which forms the opposite corner of a parallelogram
with the starting point of the two vectors as the original corner. A vector drawn
from the original corner to the opposite corner is the resultant.
Resultant
A
Equilibrant
B
B
The equilibrant is the vector needed to cancel the effects of the
Resultant.
Equilibrant + Resultant = ZERO
If the Resultant is a force, the Equilibrant is the force needed to establish
Equilibrium – no net force on the object.
Another method is tip-to-tail.
Given vectors A and B,
A
B
move A so that the tail of A is at the tip of B. Draw the Resultant from the
starting point at the tail of B to the tip of A.
Resultant
A
B
The component method is more complicated, but gives more accurate results.
Any vector can be built by adding other vectors together to make it. To
make life easy, we build a vector out of pieces pointing along the x axis and
along the y axis. If we know the angle we can find the x and y components
by using
Vector(x component) = |Vector(magnitude)| × cos(), and
Vector(y component) = |Vector(magnitude)| × sin()
Vector
Y component
 
X component
On the other hand, if we know the x and y components then we can
determine  by
𝒚
𝜽 = 𝒂𝒓𝒄 𝒕𝒂𝒏 ( )
𝒙
Also, if we know the x and y components we can use the Pythagorean theorem
to find the magnitude of the Vector
|𝑉𝑒𝑐𝑡𝑜𝑟 (𝑀𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒)| = √(𝑥 2 + 𝑦 2 )
To add two or more vectors, find the x and y components for each vector and
add them.
For our vectors A and B with A having an angle of 1 from the horizontal
axis and B having an angle of 2 above the horizontal axis we can find the
components as in the previous example.
A
1
B
Vector
Magnitude
A
B
Resultant
|A|
|B|
Equilibrant
√(𝑆𝑢𝑚 𝑜𝑓 𝑥)2 + (𝑆𝑢𝑚 𝑜𝑓 𝑦)2

Angle
X
Component
1
|A| cos (1)

|B| cos ()
𝑠𝑢𝑚 𝑜𝑓 𝑦 Sum of
𝐴𝑟𝑐𝑡𝑎𝑛
𝑠𝑢𝑚 𝑜𝑓 𝑥 above x
components
Add 180° to the
resultant
Y
Component
|A| sin (1)
|B| sin ()
Sum of
above y
components
The x and y components of the resultant give the coordinates of the end
point of the vector.