Component Method
In a plane, the intersection of two perpendicular lines, one horizontal and the other vertical, defines a reference point or origin for the plane. Each arrow in the plane is equal to the sum or resultant of a horizontal and vertical arrows, its so-called horizontal and vertical components. This representation or decomposition of an arrow as the sum of horizontal and vertical components leads to a third method for arrow addition given by the addition of components. The horizontal components of an arrow sum is given the arrow sum of the horizontal components. Likewise, the vertical components of an arrow sum is given the arrow sum of the vertical components. Here is a technical observation with little motivation except for consequences that will follow.
Each vector from the origin in a rectangular coordinate is the sum of vectors parallel to the coordinate axis's. The following diagram shows how these "component vectors" parallel to the axis's may be used instead of the parallelogram rule.
See applet for examples. Have it display rectangular coordinates for points and vectors. To learn more, visit the chapter vectors in Calculus and Beyond.
Transition to Coordinates
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.
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Complex Numbers |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the
folder.
First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
Intro assumes the field properties
of real numbers in place of
deriving them geometrically