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Online Volume 2, Three Skills for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills and concepts, those needed in calculus, again to make the hard easier. A visual understanding of complex numbers may serve as back ground info for partial fraction decomposition. |
Polar Coordinates and Scalar Multiplication
Polar Coordinates, A quick introduction
Polar coordinates (R, a) can be used to locate points [a, b] in the plane. Polar coordinates (R, a) for points [a,b] in the plane can also be measured. So rectangular and polar coordinates are interchangeable at least through measurement. Trigonometry provides further methods.
Given a point A in a planar map with coordinates, we can measure its distance R from the origin and measure in a counterclockwise manner, the angle a it makes with the horizontal coordinate axis. Geometrically or physically we assume, the distance R and the angle a uniquely determine the point, and the vector from the origin to the point
The ordered pair (r units, a ) with round brackets provides the polar coordinates of a point A in the plane. Here r units is the length of the the associated "position" vector OA which goes from the origin O to the point A. This vector makes an angle alpha with the horizontal axis.
Polar to Rectangular Coordinates, and Back
We will write
[a,b] = (R,q)
when both sides locate, determine or correspond to the same point in the plane. I assume you know how to measure the rectangular coordinates [a,b] and polar coordinates (R, q) of points in the plane, given the location. This provides a geometric mechanism for determining rectangular coordinates from polar coordinates, and vice-versa. (Methods based on trigonometry will be available later.)
The point with polar coordinates (R,q) has length R and angle q. [Angles are determined for each point, modulo 360 degrees.]
Figure 1. Rectangular and Polar Coordinates of a Point.
Each point [a,b] in the plane can be identified with the arrow, head at it, and tail at the origin. For the following topic, recall the discussion of arrow addition using horizontal and vertical coordinates.
Points in the plane may be located using polar coordinates or rectangular coordinates.
round () versus square brackets []
In the following lessons, if I remember, I will use round brackets () with polar coordinates. Your textbooks may use round brackets for both polar and rectangular coordinates. Matching pairs of round (), square brackets [] and braces {} will also be used in computations to indicate the order in which calculations are done. This could lead to bracket and parenthesis over use.
Scalar Multiplication by k > 0
First Property: If u = [a,b] = (R,q) and k > 0 then ku = k[a,b] = (kR,q) or
For a point or vector with rectangular coordinates [a, b], the product with a real number k is given by
k[a,b] = [ka, kb]
Now suppose the polar coordinates of [a,b] = (R,q) and suppose k > 0. We will find the polar coordinates of k[a,b] = [ka, kb]. By the Pythagorean theorem, the length of [ka, kb] or its distance to the origin squared is given by
(ka)2 + (kb)2 = k2a2 + k2 b2 = k2 (a2 + b2 )= k2 R2.
Therefore k[a,b] = (kR,q) in polar coordinates as scalar multiplication by k >0 does not change angles.
Second Property. Scalar Multiplication by k > 0 distributes over addition:
If u = [a, b] and v = [c,d] then
k(u+v) = k u + k v
as the left hand side = k ([a,b] + [c,d]) = k[ a+c, b+d] = [k(a+c), k(b+d)] = [ ka + kc, kb+ kd] = [ka,kb] + [kc,kd] = k[a,b] + k[c,d] = the right hand side.
The chain of equalities here was justified one a time and one after another by the definition of vector multiplication and scalar multiplication, and/or the use of the distributive law for real numbers. The argument works for k zero or positive as well, but we only need the case k > 0 below.
Instructors: Asking students to plot points [x,y] = k [1.5, -2.5] = k[a, b] in general implies angles are not changed by scalar multiplication by a variable or parameter k. Using similarity, one may argue that ka and kb as is or in magnitude provide the sides of similar right triangles. The foregoing could be illustrated for [a,b] drawn in each of the four quadrants or on a coordinate axis using numbers. Is there a simpler or more elegant way to imply k[a,b] and [a, b] will have the same direction or angle, modulo 360 degrees?
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