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Vectors are merely a representation of each coordinate in a basis direction. So for example giving directions. You could say up 2 miles, left 3 miles, and the vector would be (3,2) (assuming x,y configuration). So in x,y,z 3-space it is \[r = (x,y,z)\] or \[r = xi + yj + zk\]. So say a vectors travels right 2 units, up 3 units, and forward 4 units, the vector r is : \[r = (2, 3, 4)\]
by doing this you can then add vectors quite simply by merely adding their respective components. \[r = (2,3,4) \] and \[v = (1, -3, 6) \]\[r+v = (2+1, 3-3, 4+6) = (3, 0, 10)\]
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also, one thing, don't confuse vectors with coordinates(ie points) a vector has a length and a direction. a point is just a location in space. So for example I could place a vector \[v = (2,2,2)\]on a point \[P(1,1,1)\] and this vector would if I were to say follow it, would then point to the point say \[Q(1+2,1+2,1+2) = Q(3,3,3)\] so remeber that a vector is NOT a point, it describes both a direction and a magnitude. BTW the magnitude of a vectore is : \[|v| = \sqrt(2^2 + 2^2 + 2^2 ) = \sqrt(12) = 2\sqrt(2)\] this is the magnitude or "norm" of a vector