## Zenmo one year ago 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

1. Zenmo

Parametric Equations of a Line in Space: $x = x _{1}+at, y = y _{1}+bt, and z= z _{1}+ct$

2. 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

3. Michele_Laino

so we have: $\Large x\left( t \right) = - 4 + \frac{t}{2}$ similarly for y(t) and z(t)

4. Zenmo

$< -4, 1, 0 > + t <\frac{ 1 }{ 2 }, \frac{ 4 }{ 3 }, -1>$

5. Zenmo

=$<-4+\frac{ 1 }{ 2 }t, 1+\frac{ 4 }{ 3 }t, -t>$ Is that correct for parametric equations?

6. Zenmo

@Michele_Laino

7. Zenmo

The solution is: x = -4 + 3t, y= 1+8t, z = -6t. It doesn't make sense?

8. Michele_Laino

9. Michele_Laino

other parametric equations, which are equivalent to the first ones, are: $\Large \left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 4} \\ 1 \\ 0 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 3 \\ 8 \\ { - 6} \end{array}} \right)$

10. Michele_Laino

so you are right!

11. 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.

12. Zenmo

so its x= -4t +3, y= 1+8t, z=-6t

13. Michele_Laino

it suffice that you change your parameter, namely: $\Large t \to 6\tau$ where \tau is the new parameter

14. Michele_Laino

$\Large \left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 4} \\ 1 \\ 0 \end{array}} \right) + \tau \left( {\begin{array}{*{20}{c}} 3 \\ 8 \\ { - 6} \end{array}} \right)$

15. Zenmo

I need a slight help on another problem on converting into symmetric equations, I already did the parametric part.

16. Michele_Laino

ok! I see your new problem