Vector Addition - Algebra

Adding points

When we add the coordinates of two POINTS a and b we see that we get the coordinates of a point c which is the vertex of a parallelogram, with a and b being two adjacent vertices and the origin O being the other.

Algebraic method of adding vectors

To add VECTORS, suppose that the tails of V and W are at the origin and their heads are at points a and b. Then if the tail of V+W is at the origin, the head of V+W is at the point whose coordinates are the sum of those of a and b.

Given the parallelogram with vertices O, a, b, c, addition of point coordinates shows that c = a + b. But V has coordinates a, W has coordinates b, and V+W has coordinates c = a + b. What if the tails of V or W are not at the origin? What then?

It doesn't matter. Move V and W by parallel displacement so that tails are at origin. V and W are the same vectors. They have coordinates a and b, no matter where they are.

Moral: Addition can either be done geometrically by constructing a parallelogram, or algebraically by adding the components of a and b. The key thing here is that no matter where V and W have wandered to, there are copies of them with tails at the origin. THESE copies determine the points corresponding to the vectors, and its the sum of the coordinates of THESE points that determine the point corresponding to V+W.

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