Subtraction Geometrically

de-CH

utf-8

math math-format graphie

komplex-01

custom

10000

randRangeNonZero(-5, 5) randRangeNonZero(-5, 5) randRangeNonZero(-5, 5) randRangeExclude(-5, 5,[-COMPLEX1])

Let {\color{blue}z} and {\color{red}w} complex numbers.

Move the point at the origin to the difference {\color{blue}z} - {\color{red}w}.

graphInit({ range: 11, scale: 20, tickStep: 2, labelStep: 1, }); label([ 11, 1], "\\operatorname{Re}", "left"); label([ 0.5, 10], "\\operatorname{Im}", "right"); line([0, 0], [REAL1, COMPLEX1], { stroke: BLUE, arrows: "->" }); line([0, 0], [REAL2, COMPLEX2], { stroke: RED, arrows: "->" }); var AF = 1 + 0.8 / sqrt(REAL1 * REAL1 + COMPLEX1 * COMPLEX1); label([AF * REAL1, AF * COMPLEX1], "z", { color: BLUE }); var BF = 1 + 0.8 / sqrt(REAL2 * REAL2 + COMPLEX2 * COMPLEX2); label([BF * REAL2, BF * COMPLEX2], "w", { color: RED }); addMouseLayer(); graph.guessPoint = addMovablePoint({ constraints: {}, snapX: 0.5, snapY: 0.5, });

[ graph.guessPoint.coord ]

return ((guess[0][0] === REAL1 - REAL2) && (guess[0][1] === COMPLEX1 - COMPLEX2));

graph.point.setCoord(guess);

We first do the subtraction geometrically above.

line([REAL1, COMPLEX1], [REAL1 - REAL2, COMPLEX1 - COMPLEX2], { stroke: RED, arrows: "->" }); graph.guessPoint.toFront();

line([0, 0], [REAL1 - REAL2, COMPLEX1 - COMPLEX2], { stroke: ORANGE, arrows: "->" }); graph.guessPoint.toFront(); graph.guessPoint.moveTo(REAL1 - REAL2, COMPLEX1 - COMPLEX2);

Formally, we obtain the coordinates by subtracting the respective real and imaginary parts.

We see above that {\color{blue}z} = REAL1 + coefficient(COMPLEX1)i and {\color{red}w} = REAL2 + coefficient(COMPLEX2)i :

z - w = (\purple{REAL1} \pink{ + coefficient(COMPLEX1)i}) - (\purple{REAL2} \pink{ + coefficient(COMPLEX2)i})

\hphantom{z - w} = \purple{(REAL1 - REAL2)} + \pink{(COMPLEX1 - COMPLEX2)i}

\hphantom{z + w} = \purple{REAL1 - REAL2} \pink{ + coefficient(COMPLEX1 - COMPLEX2)i}.