Science:Math Exam Resources/Courses/MATH200/December 2013/Question 01 (a) i/Solution 1

We want to find the symmetric equations for L where:

$L={\frac {x-x_{0}}{a}}={\frac {y-y_{0}}{b}}={\frac {z-z_{0}}{c}}$

We can find the constants in the above equation by arranging the vector parametric equation given in the question into the form of:

$r(t)=\langle x_{0},y_{0},z_{0}\rangle +t\langle a,b,c\rangle$

Expanding the equation given, we get:

$r(t)=2\mathbf {i} +3t\mathbf {i} +0\mathbf {j} +4t\mathbf {j} +-1\mathbf {k} +0t\mathbf {k}$

Rearranging:

$r(t)=(2\mathbf {i} +0\mathbf {j} -1\mathbf {k} )+t(3\mathbf {i} +4\mathbf {j} +0\mathbf {k} ):$

Which is simply:

$r(t)=\langle 2,0,-1\rangle +t\langle 3,4,0\rangle$

where

$\displaystyle x_{0}=2,y_{0}=0,z_{0}=-1,a=3,b=4,c=0.$

Plugging them into the symmetric equation, we get:

$L={\frac {x-2}{3}}={\frac {y-0}{4}}={\frac {z-(-1)}{0}}$

Because we cannot have zero in the denominator, our final answer is:

${\color {blue}L={\frac {x-2}{3}}={\frac {y}{4}},\quad z=-1.}$