**Symetric-form Equations, Direction Numbers, and Parametric Equations**

Straight lines in Cartesian three-space present a more complicated picture than straight lines in the Cartesian coordinate plane. This is because there is an added dimension, making the expression of the direction more complex. But all linear equations, no matter what the number of dimensions, have one thing in common: they can be reduced to a form where no variable is raised to any power other than 0 or 1.

**Symmetric-form Equation**

A straight line in Cartesian three-space can be represented by a “three-way” equation in three variables. This equation is known as a *symmetric-form* *equation* . It takes the following form, where *x* , *y* , and *z* are the variables, ( *x* _{0} , *y* _{0} , *z* _{0} ) represents the coordinates of a specific point on the line, and *a* , *b* , and *c* are constants:

( *x* − *x* _{0} )/ *a* = ( *y* − *y* _{0} )/ *b* = ( *z* − *z* _{0} )/ *c*

This requires that none of the three constants *a* , *b* , or *c* be equal to zero. If *a* = 0 or *b* = 0 or *c* = 0, the result is a zero denominator in one of the expressions, and division by zero is not defined.

**Direction Numbers**

In the symmetric-form equation of a straight line, the constants *a* , *b* , and *c* are known as the *direction numbers* . If we consider a vector m with its end point at the origin and its “arrowed end” at the point ( *x,y,z* ) = ( *a,b,c* ), then the vector **m** is parallel to the line denoted by the symmetric-form equation. We have:

**m** = *a* **i** + *b* **j** + *c* **k**

where **m** is the three-dimensional equivalent of the slope of a line in the Cartesian plane. This is shown in Fig. 9-12 for a line *L* containing a point *P* = ( *x* _{0} , *y* _{0} , *z* _{0} ).

**Parametric Equations**

There are infinitely many vectors that can satisfy the requirement for **m** . If *t* is any nonzero real number, then *t* **m** = ( *ta,tb,tc* ) = *ta* **i** + *tb* **j** + *tc* **k** will work just as well as **m** for the purpose of defining the direction of a line *L* . This gives us an alternative form for the equation of a line in Cartesian three-space:

*x* = *x* _{0} + *at*

*y* = *y* _{0} + *bt*

*z* = *z* _{0} + *ct*

The nonzero real number *t* is called a *parameter* , and the above set of equations is known as a set of *parametric equations* for a straight line in *xyz* -space. In order for an entire line (straight, and infinitely long) to be defined on this basis of parametric equations, the parameter *t* must be allowed to range over the entire set of real numbers, including zero.

**Lines with 3D Coordinates Practice Problems**

**PROBLEM 1**

Find the symmetric-form equation for the line *L* shown in Fig. 9-13.

**SOLUTION 1**

The line *L* passes through the point *P* = (−5,−4,3) and is parallel to the vector **m** = 3 **i** + 5 **j** − 2 **k** . The direction numbers of *L* are the coefficients of the vector **m** , that is:

*a* = 3

*b* = 5

*c* = −2

We are given a point *P* on *L* such that:

*x* _{0} = −5

*y* _{0} = −4

*z* _{0} = 3

Plugging these values into the general symmetric-form equation for a line in Cartesian three-space gives us this:

( *x* − *x* _{0} )/ *a* = ( *y* − *y* _{0} )/ *b* = ( *z* − *z* _{0} )/ *c*

*x* − (−5)]/3 = [ *y* − (−4)]/5 = ( *z* − 3)/(−2)

( *x* + 5)/3 = ( *y* + 4)/5 = ( *z* − 3)/(−2)

**PROBLEM 2**

Find a set of parametric equations for the line *L* shown in Fig. 9-13.

**SOLUTION 2**

This involves nothing more than rearranging the values of *x* _{0} , *y* _{0} , *z* _{0} , *a* , *b* , and *c* in the symmetric-form equation, and rewriting the data in the form of parametric equations. The results are:

*x* = −5 + 3 *t*

*y* = −4 + 5 *t*

*z* = 3 − 2 *t*

Practice problems for these concepts can be found at: Vectors And Cartesian Three-Space Practice Test.

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