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 .

is also perpendicular.

Approach

The two lines and are perpendicular if the equation 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}

Formulate the perpendicular condition for the second pair of lines.

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

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]]}