A vector is a group of numbers. For example, (2, 3, 1) is a vector. Vectors that have the same quantity of numbers can be added by adding their components (1, 2, 3) + (4, 5, 6) = (5, 7, 9). They can be also multiplied by a number by multiplying each individual component: 3*(1, 2, 3) = (3, 6, 9).
Here you see two green vectors (1.5, 2.5) and (0, 2) added to form a red vector (1.5, 4.5). As you can see, order of addition does not matter.
I have a pretty sophisticated Linear Algebra Workbench that can do vector operations and much more.
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