Vector Addition:
Adding two vectors produces a third vector, shown geometrically as follows.
Given vectors u
and v
1. Sketch the vectors so that the second vector, v, so is “head to tail” with u
v
u
2. Then draw the new vector, w, starting from the tail of u and extending to the head of v.
v
u
w= u+ v
(resultant vector)
Notice that vector addition is commutative, i.e. that u + v = v + u
v
u
w=u+v=v+u
u
v
The two paths u + v and v + u take you to the same endpoint. This diagram also illustrates the
“parallelogram rule” for addition of vectors, where the sum of the two vectors is the diagonal of the
parallelogram formed by the two vectors.
Vector Subtraction: Subtracting two vectors, such as u – v or v – u, also produces a new vector.
Given vectors u
and v
1. Sketch the vectors originating from the same point, i.e. “tail to tail”.
2. Draw the vector from the head of the second vector to the head of the first.
u
u
w1 = u – v
w2 = v – u
v
v
Note:
1. Subtraction is not commutative. u – v is not the same as v – u. They are opposites as we would
expect, shown above by the two vectors w1 and w2 which have the same length but opposite
directions.
2. in general the “difference” between 2 mathematical objects (such as numbers, vectors, matrices,
functions, etc) is the new object which must be added to the second to obtain the first. In the first case
above (u – v) + v = u and in the second case (v – u) + u = v.
Alternatively, Add the Opposite: Rewrite u – v as u + (-v) and use the addition rule:
v
-v
w = u + (-v)
=u–v
the resultant vector here is the same as w1 above
u
Applications:
1. The Horizontal and Vertical components of a vector:
v
vy
v = vx + v y
θ
vx
We consider two problem types
a. Given
| v | and θ,
b. Given | vx | and | vy |,
find | vx | and | v y |
find | v | and θ
2. Static Equilibrium:
The sum of the force vectors acting on an object gives the net, or combined, force. If an object is at rest, not
moving, then the net forces acting on it must be zero, or more precisely, the “zero vector” which has a
magnitude of zero and no direction.
Suppose we have three force vectors, F1, F2, and F3. Form the sum, F1 + F2 + F3, using the “tail to head”
method shown above.
Case 1: Here is a case where the net force is not the zero vector (because the force vectors don’t form a
closed path). This object would be moving in the direction indicated by the sum of the vectors.
F1 + F2 + F3
F3
F1
F2
Case 2: Here is a case where the net force is the zero vector, so this object would be in static equilibrium.
F3
F1
F2
F1 + F2 + F3 = 0 in this case because when the force vectors form a closed path their sum has no
length or direction, there is no arrow to draw for the resultant vector.