The symmetric equation of a line is an alternate way to express a distinct line in R3 or higher. In R2 there are easier ways of writing it.

A symmetric equation of a line is most often formed from the parametric equation of a line, and is sightly more compact.

The symmetric equation of a line sets each component equal to the common parameter, and then sets each component equal to every other one. Before continuing, you may wish to review the vector equation of a line and the parametric equation of a line.

The general form of the parametric equation of a line is:


(Please note that the terms are generally expressed as fractions rather than plain divisions.)

To form this equation, treat each expression in the parametric equation of a line as an equality (for example, in the parametric equation, you have

x: ax + (t)(m1)
It should be treated as if it was
x=ax + (t)(m1)
Solve each expression for the parameter, t. Since each of the resultant expressions is equal to the constant t, they are also equal to each other. Express them this way.

It is interesting to note that this expression of a line breaks one of the more common tenets of good form in most mathematical writings, which to limit yourself to one equality sign per line. Here, there are always one fewer equality signs for the line than there are dimensions in the line.

It is also possible to use this method of expressing a line in fewer or more dimensions. For example, if this form were to be used in R2, the z expression would simply be omitted (there would only be one equality sign). If you wanted to express a line in R3 that was parallel to a coordinate plane, the term for the axis to which it was parallel is also omitted. For example, if the line was parallel to the x-z plane, you would end up with an expression with only a z and x term. The extra information would be found at the side, with a standalone expression for the y term (y=k, where k is some constant.)

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