If necessary, review:

**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*

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