ES-10 Two Dimensional Vectors - Addition
Vectors possess magnitude and direction.
Graphically a vector is represented by an arrow, the length of the arrow proportional to the magnitude of the vector and the head of the arrow indicating the direction.
The vector shown has been resolved into x and y components.


If the direction of the vector is indicated with respect to the x axis,



tan
From trigonometry we can determine the magnitude of the vector using the Pythagorean Theorem, as well the angle from a major axis.


Two or more vectors may be added graphically (Parallelogram Law) or analytically by using components.




For example, we have two vectors given in terms of magnitude and direction as shown here. They may be added graphically by drawing a vector parallel to A through the head of B and a vector parallel to B through the head of A. The diagonal connecting the original junction of the tails of A and B and the opposite corner is the vector sum of A and B.

C represents the vector addition of A and B.


Alternately, the vectors may be added analytically by the following method:

A = 5 cos(25)i + 5 sin(25) j
A = 4.53 i + 2.11 j

B = 4 cos(65) i + 4 sin(65) j
B = 1.69 i + 3.62 j
1. Resolve both vectors into x and y components where i and j are unit vectors (vectors of magnitude 1) in the x and y directions, respectively.



C = A + B
C = (4.53+1.69)i + (2.11+3.62)j
C = 6.22 i + 5.73 j
2. Add like components




3. Determine magnitude and direction of C if needed.

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