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Lines with 3D Coordinates Practice Problems

— McGraw-Hill Professional
Updated on Oct 3, 2011

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.

 

Vectors and Cartesian Three-Space Straight Lines Parametric Equations

Fig. 9-13 . Illustration for Problems 9-9 and 9-10.

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:

( xx 0 )/ a = ( yy 0 )/ b = ( zz 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.

Vectors and Cartesian Three-Space Straight Lines Parametric Equations

Fig. 9-13 . Illustration for Problems 9-9 and 9-10.

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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