College Preparatory Program • Saudi Aramco
Addition and Subtraction of Vectors
Addition and Subtraction of Vectors
ADDITION:
The rules for adding vectors are best understood using diagrams. To add vectors A and
B, firstly draw the vectors A and B. A scale must be used by which the size of the vectors will be
represented by the length of the arrow as in Fig 1.
B
Fig 1
A
They are then joined as in Fig 2. The method is sometimes called “tail to tip“ in that the
tail of the second vector is applied to the tip of the first one or vice versa.
Fig 2
B
A
The resultant: R = A + B can be drawn as the third side of a triangle, from the tail of the
first vector to the tip of the second one as in Fig 3.
Fig 3
R
B
A
Using diagrams, it can be easily shown that vector addition is commutative: A + B = B + A
A
B
β
θ
α
Vector addition is also associative:
R
A + (B + C) = (A + B) + C
Because when adding two vectors diagrammatically you form a triangle (sometimes
called the triangle of forces) you could find the magnitude and direction of the resultant by
using:
College Preparatory Program • Saudi Aramco
Addition and Subtraction of Vectors
–
1. The Law of Cosines:
2. The Law of Sines:
=
, followed by
=
However the most favored method, especially when adding more than two vectors is to
use addition of components. First each vector must be resolved into its horizontal and vertical
components.
e.g.
A Sin θ
A
A
A Cos θ
θ
With two or more vectors the horizontal ( x ) and vertical ( y ) components are added.
Rx = Ax + Bx + Cx
and
Ry = Ay + By + Cy
The magnitude of the resultant is achieved by the Theorem of Pythagoras and the
direction by the tan-1 of the angle.
Θ = tan-1 ( )
Example: Add the following vectors.
A = 25
B=4
300
850
C=6
500
College Preparatory Program • Saudi Aramco
Addition and Subtraction of Vectors
Ax = A cos 500, Bx = B cos 1000, Cx = C cos 2650
Ay = A sin 500 , By = B sin 1000 , Cy = C sin 2650
Horizontal addition: Rx = 25 cos 500 + 4 cos 1500 + 6 cos 2650 = 12.08
Vertical addition:
Ry = 25 sin 500 + 4 sin 1500 + 6 sin 2650 = 15.17
Resultant magnitude =
= 19.39
Resultant direction = tan-1 (
) = 51.470
SUBTRACTION:
There are two methods of subtraction. The first is to regard subtraction as the addition of the
negative of the vector to be added i.e. R = A – B is the same as R = A + (- B).
e.g.
B
A
-B
R=A-B
The second method:
1. Draw the two vectors from the origin or the same point.
B
A
2. Draw R, the difference, from the head of B to the head of A as in the diagram.
B
R=A-B
A