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

MATH200 December 2013

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Question 01 (a) i
The line $\displaystyle L$ has vector parametric equation $\displaystyle {\textbf {r}}(t)=(2+3t){\textbf {i}}+4t{\textbf {j}}-{\textbf {k}}$ i. Write the symmetric equations for $\displaystyle L$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Make this equation into a more general vector parametric equation that you may be more familiar with: $r(t)=\langle x_{0},y_{0},z_{0}\rangle +t\langle a,b,c\rangle .$

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Solution 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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: ${\color {blue}L={\frac {x-2}{3}}={\frac {y}{4}},\quad z=-1.}$

Solution 2
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. To be deleted..