Exploring Analyic Geometry with Mathematica®

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Equations of Perpendicular Lines

lnsperp.html

Exploration

Show that the pair of lines a x+b y +c=0 and b x-a y+c'=0 are perpendicular. Show that the pair

a x+b y+c=0 and "lnsperp_1.gif".

is also perpendicular.

Approach

The two lines "lnsperp_2.gif" and "lnsperp_3.gif" are perpendicular if the equation "lnsperp_4.gif""lnsperp_5.gif" is true. The two pairs of lines given can be shown to be perpendicular by examining this equation.

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

Formulate the perpendicular condition for the first pair of lines.

Clear[A1,B1,A2,B2,a,b];
A1*A2+B1*B2 /. {
   A1->a, B1->b, A2->b, B2->-a}

"lnsperp_6.gif"

Formulate the perpendicular condition for the second pair of lines.

A1*A2+B1*B2 /. {
   A1->a, B1->b, A2->1/a, B2->-1/b}

"lnsperp_7.gif"

Discussion

Notice that the second pair of lines can be derived from the first by dividing the first equation by the quantity a b. However, this is invalid if either a or b is zero. The relationship shown for the first pair is valid for all lines.  The Descarta2D function IsPerpendicular2D also verifies that the pairs are perpendicular.

Clear[c1];
{IsPerpendicular2D[Line2D[a,b,c],Line2D[b,-a,c1]],
IsPerpendicular2D[Line2D[a,b,c],Line2D[1/a,-1/b,c1]]}

"lnsperp_8.gif"


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