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Home < Geometry - maps plans trigonometry vectors < 13 Vectors << 12 From Applied To Pure Mathematics
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From Applied to Pure MathematicsIn pure mathematics, arrows and vectors are not drawn. They are only appear as coordinates. But in applied mathematics, we assume those coordinates correspond to vectors that can be drawn, multiplied by scalars and added graphically. The explanations below show how the assumption of the parallelogram method leads to component and coordinate method for adding vectors. Coordinate View (Again)If [a,b] and [c,d] are the heads of two vectors, we the head of the sum of the vectors will be at location [e, f] = [a,b] +[c,d] where the + operation indicates arrow addition. In the case where both heads are in the first quadrant, we have e = a +c and f = b +d. In the case where one or both are not in the first quadrant, put a + c = e and b + d = f. This defines the addition of coordinates. Exercise: Show this addition is well-defined. Observe, addition of vectors using the first method (parallelogram method) commutes. That the order of addition does not affect the result. Therefore addition commutes in all further methods that produce the same result. This implies the addition of coordinates just defined commutes Coordinate Addition Method
The next figure shows each vector as a sum of components. We assume the vector heads are located at [a,b] and [c,d]. Each vector from the origin may be defined (or drawn) by giving the location of its head.
The next step is to arrive at a formula for vector addition in terms of the coordinates [a,b] and [c,d] of the head locations of the vectors or summands in this vector addition.
The foregoing suggests we represent points in the plane and vectors ending at those points by the coordinates of the head of the vector, that is the coordinates of the points. The foregoing also suggests the vector sum of the arrows with heads at [a,b] and [c,d] respectively has its head at [a+c, b+d]. So to represent the vector addition of arrows, we put [a+c, b+d] = [a,b] + [c,d] This gives the coordinate way to add points and their position vectors in the plane. The position vector of a point goes from the origin (its tail) to the point (its head). |
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Home < Geometry - maps plans trigonometry vectors < 13 Vectors << 12 From Applied To Pure Mathematics
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13][14] [15] [16] [17]
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