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Plane Graphic Calculator from J.E.Lillge

One of the wonders of planegraph is, that it is a powerful2D numerical calculatortoo. Click on "Detach" and on "Details". Enter into the f-field of planegraph any numbers and vectors and 2x2-matrices and nearly every operation you can think of - and hit the Enter-key on your keyboard. The result is shown in the f-field of "Details".

And another wonder of planegraph is, that you can calculate inmixed modetoo, just like my "Forget C, it's all R2, it's all 2D- calculator". So Planegraph is the v e r y f i r s t, who has this property. Type: i * (3 , 4) or: ( - 5 + 4 * i) * ( 4.1 , 3 ) And You can calculate sin (2,5) or (3+4i)dot(6+9i),..

When You enter into f a formula or a term of a, You can move a on the plane, and the dependent f-vector has to follow, both displayed as locations, positions, as arrows fixed to the origin. You want to display a function f=f(a) asvector field? "Here is an example showing how you can enter your vector field for (a^2+sin a) into the calculator: g = a+(a^2+sin a)(t>0) (You will probably want to make sure that the "Line" checkbox is not checked.) As you can see, the vector is drawn from the tip of a to the tip of a+f, where: f = (a^2+sin a) --------------------------------------------- The general form for displaying a vector field in the calculator is: g = a+(f)(t>0) where (f) is replaced by any parenthesized vector function of a." J.E.Lillge wrote to me. So now we can move the vector a on the plane like a probe investigating this field, and the vector f will follow.

This feature gives the beautiful possibility to## Teach the basics of a geometric vectorspace

Use the keyboard and let the mouse to Your pupil. Click on Detach and on Detail, click on the three boxes Plot, Spin, Line until they are all blank. Change in the box function to: user-defined. Enter a vector 1+2i or (1,2) into m by typing a + ((1+2i)) (t>0) (Enter) a denotes the tail of an arrow and (t>0) draws it Your pupil will see: an arrow - with a very sharp tip. He can grab it by the tail, the point a and move it around. A vector is geomtrically an arrow, a straight line-segment with a direction.The direction relates to the environment and/or the other arrows. By translating the arrow, you don't change it. Enter a second vector (1.5, - 0.7) into n: b + ((1.5, - 0.7)) (t>0). Into f You'll always enter Your calculation, here "a+B" or "a-b": c + ((1+2i) + (1.5, - 0.7)) (t>0). So the pupil can grasp the addition of arrows. Of course You will show him some different vectors too. In geometry You have learned, how to magnify/minimize or resize a straight line-segment: attach a second line at an angle betweeen 30 to 60 degrees to it, mark with compasses a unit and any length, let's say three times the unit, on it. (With compasses You can devide the unit in the middle, the parts again in their middle, and by repeating this, you can get the part behind the decimal point of any real number, in a binary form). Join the unit-mark on this "scale" with the end of the straight line-segment and draw a parallel through the 3. - mark on the "scale". Where this meets the prolonged line-segment is the end of the resized line-segment. Enter into n: b + ((3.5 * (1,2)) (t>0) b displays the 3.5 - fold of a, it's a multiplication of an arrow by a number, called: real scalar multiplication (r.s.m.). Addition and r.s.m. together makes the set of directed line-segments or arrows on a plane to a geometric vectorspace. You can calculate and display linear combinations of arrows. Enter into n: b + ((1.5, - 0.7)) (t>0) Enter into f: c + ((2*(1+2i) - 0.5 *(1.5, - 0.7)) (t>0). It's how Wessel started in the old-greek geometry style, without the underlying grid, without a coordinate system, an origin, a scale or a unit, without a pricipal or basic direction. And it's good not only for 2D, but for 3D too, and not only on a paper. Do it in reality with the pupil: Go three paces north, turnleft and go another seven paces, six steps up the stairs ,..

## ...the geometric Bombelli vectorspace

We add a multiplication (2D only). In effect it's a rotation (addition of angles) and a stretching/shrinking (multiplication of length). It's only 2D, because on a plane all axes of rotation are perpendicular (or normal) to the plane - thus only parallel axes are used. (To my best knowledge the only known representation of a 2D - vectorspace with multiplication like this is a plane with it's arrows.) It's called a multiplication, because it's one extension of the normal multiplication of (real) numbers, just like the addition of arrows contains the addition of length on one line, id est numbers. We didn't represent numbers on the plane up to now, but we did use them as factors in the r.s.m. The common geometric representation of a quantity, a number is a straight and sometimes a circle line-segment. And a product is an area, a rectangle , but not a line-segment. So You can't multiply two line-segments, unless you introduce a unit or a scale. Enter into m: a + ((1)) (t>0) and into n: b + ((1.5)) (t>0) and into f: c + ((dx*(1)) * (dx *(1.5)))(t>0) You see how the product "a times b" changes, when You change the length of the unit d by moving d horizontal. The multiplication of two straight line-segments of length in terms of a unit is done geometrically somehow similar to the r.s.m.:draw two line-segments at an angle to each other, mark on one the unit and connect this mark to the tip of the other. Draw a parallel through the tip of the first and this meets the prolonged second line segment in the tip of the resulting line-segment. You can not multiply directions. In the long math-lessons i often wondered, that multiplication of vectors is represented by rotations, but they never told, what it is, what they rotated. Let's start with difference of direction, with division (that's how Hamilton started with Quaternions). From north-east to north you get the same difference of direction, as from -let's say - south-west to south, 1/8 th of a complete circle - and these rotations are orientated both anticlockwise or with left-turn. And we will stay with this orientation left turning, so a difference from south to southwest turning clockwise is not admitted here. Of course You can rotate from south to south-west anticlockwise, but the angle is nearly a full circle (the conjugate angle for south-west to south). Difference of direction - that's called an angle. Oriented (here:left-turn) difference of direction - that's an oriented angle. So now we disentangle arrow b from the environment and the direction of the other vectors. Enter into m: a + ((1.5,1)) (t>0) and into n the arrow (2,1.4) or 2+i*1.4 by: b + d/|d| *((2,1.4))(t>0) + d/|d| *( - (2,1.4))(t>1) + d/|d| *((cos t, sin t))(t>6.28) and into Parametric value for t: t from 0 to 6.89 as (6.89=2*pi + 35/360 * 2*pi) and into f a zero. If You want to display the unit enter into f: c + (1)(t>0), As You see, the reference-direction and the arc associated with b has a radius of one unit. Move b by its tail and by changing d on the plane. An arrow with an arc of zero degrees (t from 0 to 6.28) is a line segment, a length without direction. Finally we can multiply a * b, a rotation of the direction of a by the arc of b and with orientation left-turn and the length of both arrows multiplied with regard to the unit: Enter into f: c + ((1.5,1) * (2, 1.4)) (t>0). When You proceed to multiply or divide two rotations, the best is to open up planegraph a second time, so You get four more indepent handles (variables) and one window extra to display the result. That you can't multiply two directions is not specific for 2D or this multiplication. You can't add house-number 3 to house-number 7, or add two dates.They are called ordinal-numbers. By walking from house-number 3 a kardinal-number of 4 houses on you get to house-number 7.## ....the analytical representation of arrows as vectors

that's, what You have done all the time by entering numbers and vectors. They relate to the underlying grid. So You introduce an origin, a pricipal or basic direction - in short a coordinate-system. You will notice differences between an algebraic and the geometric representation of a vector-space. You can check on the definitions, for example: For every arrow exists an (opposite) arrow and they add up to zero, a point without direction. And You can proceed to the projection of an arrow onto the axes, the dot-product and the Euclid-vector-space. May be You have noticed, that the axes of the rotations are not fixed to a point on the plane: a ship can take aturnleft around a fire-house, the axis or the mast is moving and the ship and the direction of travelrotates.