Vectors are manipulated somewhat differently. The direction must always be taken into account when manipulating them.
When you add two vectors, it is as if you took the vectors and placed them end to end, and created a new vector running from the starting point to the end point, as demonstrated in the picture to the right. If the vectors have the same direction, then this just means adding the magnitudes, but if they have different directions, it can become more complex.
You add vectors by breaking them into their components and then adding the components, as below:
a + b = c
ax + ay + bx + by =
(ax + bx) + (ay + by) = cx + cy
The two x-components will result in the x-component of the new variable, while the two y-components result in the y-component of the new variable.
Properties of Vector Addition
The order in which you add the vectors does not matter (as demonstrated in the picture). In fact, several properties from scalar addition hold for vector addition:
Identity Property of Vector Addition
a + 0 = a
Inverse Property of Vector Addition
a + -a = a - a = 0
Reflective Property of Vector Addition
a = a
Commutative Property of Vector Addition
a + b = b + a
Associative Property of Vector Addition
(a + b) + c = a + (b + c)
Transitive Property of Vector Addition
If a = b and c = b, then a = c