Zenmo
  • Zenmo
Find a set of (a) parametric equations and (b) symmetric equations for the line through the point and parallel to the specified vector or line. (For each line, write the direction numbers as integers.) Point ( -4, 1, 0), Parallel to v = (1/2)i + (4/3)j - k
Mathematics
chestercat
  • chestercat
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Zenmo
  • Zenmo
Parametric Equations of a Line in Space: \[x = x _{1}+at, y = y _{1}+bt, and z= z _{1}+ct\]
Michele_Laino
  • Michele_Laino
for part A, we have to apply this eqaution: \[\Large X = A + tv\] whic, can be rewritten by components, like below: \[\Large \left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 4} \\ 1 \\ 0 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} {1/2} \\ {4/3} \\ { - 1} \end{array}} \right)\] where t is the parameter
Michele_Laino
  • Michele_Laino
so we have: \[\Large x\left( t \right) = - 4 + \frac{t}{2}\] similarly for y(t) and z(t)

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Zenmo
  • Zenmo
\[< -4, 1, 0 > + t <\frac{ 1 }{ 2 }, \frac{ 4 }{ 3 }, -1>\]
Zenmo
  • Zenmo
=\[<-4+\frac{ 1 }{ 2 }t, 1+\frac{ 4 }{ 3 }t, -t>\] Is that correct for parametric equations?
Zenmo
  • Zenmo
Zenmo
  • Zenmo
The solution is: x = -4 + 3t, y= 1+8t, z = -6t. It doesn't make sense?
Michele_Laino
  • Michele_Laino
your parametric equations are right!
Michele_Laino
  • Michele_Laino
other parametric equations, which are equivalent to the first ones, are: \[\Large \left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 4} \\ 1 \\ 0 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 3 \\ 8 \\ { - 6} \end{array}} \right)\]
Michele_Laino
  • Michele_Laino
so you are right!
Zenmo
  • Zenmo
Yea, I figured it out, the book just formatted the answer differently by multiplying 6 to each T to get rid of the fractions.
Zenmo
  • Zenmo
so its x= -4t +3, y= 1+8t, z=-6t
Michele_Laino
  • Michele_Laino
it suffice that you change your parameter, namely: \[\Large t \to 6\tau \] where \tau is the new parameter
Michele_Laino
  • Michele_Laino
\[\Large \left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 4} \\ 1 \\ 0 \end{array}} \right) + \tau \left( {\begin{array}{*{20}{c}} 3 \\ 8 \\ { - 6} \end{array}} \right)\]
Zenmo
  • Zenmo
I need a slight help on another problem on converting into symmetric equations, I already did the parametric part.
Michele_Laino
  • Michele_Laino
ok! I see your new problem

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