Exploring Analyic Geometry with Mathematica®

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Line General Equation Determinant

lndet.html

Exploration

Show that the general equation of a line A x+B y+C=0 is coincident with the line

"lndet_1.gif"

given in determinant form.

Approach

Evaluate the determinant and show that the result is equivalent to the equation of the specified line.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Use the Det command to form the determinant.

Clear[x,y,A1,B1,C1];
d=Det[{{x,y,1},
       {-A1*C1,-B1*C1,A1^2+B1^2},
       {B1,-A1,0}}] //Simplify

"lndet_2.gif"

The line represented by the determinant is coincident to the given line.

Line2D[d,{x,y}] //Simplify

"lndet_3.gif"

Discussion

If the constant coefficient of the line is non-zero (C≠0) then a simpler determinant represents the line and is given by

"lndet_4.gif".

d=Det[{{x,y,1},{-C1,0,A1},{0,-C1,B1}}]

"lndet_5.gif"

Line2D[d,{x,y}] //Simplify

"lndet_6.gif"


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