## theanonymous27 Group Title Find a parametrization r(t) = h x(t), y(t), z(t)i of the straight line passing through the origin in 3-space whose projection on the xy-plane is a line with slope 4, while its projection on the yz-plane is a line with slope −3, i.e., ∆y/∆x= 4,∆z/∆y= −3. one year ago one year ago

1. wio Group Title

Can you come up with a vector which goes in the correct direction?

2. theanonymous27 Group Title

How is that?

3. wio Group Title

First, it doesn't matter what the magnitude of the vector is... so we get to chose one of the components as long as the others are correctly proportional.

4. wio Group Title

Our vector only has to have the right direction.

5. wio Group Title

@theanonymous27 Does this make sense?

6. theanonymous27 Group Title

I understand, but how can I find that vector? what I dont get is how can I use the slopes here to find what I need

7. wio Group Title

So we can start by letting our $$x$$ component be 1:$\mathbf{v} = \begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix} = \begin{bmatrix} 1 \\ y_1 \\ z_1 \end{bmatrix}$

8. wio Group Title

Since $$\Delta y/\Delta x= 4$$ and $$\Delta x = 1$$, what must $$\Delta y$$ be?

9. wio Group Title

We use the slopes to figure out the proportions between each component, @theanonymous27

10. theanonymous27 Group Title

so <Y is 4, and then we get that Z has to be -12

11. wio Group Title

Yes

12. theanonymous27 Group Title

so does the slope represent the component of the vector?

13. theanonymous27 Group Title

we end up with r(t) = <t, 4, -12t> ? is that right?

14. wio Group Title

So given the direction vector $$\mathbf{v}$$ and some point which the line goes though $$\mathbf{b}$$ the parametrization for our line is: $\mathbf{r}(t) = \mathbf{v}t+\mathbf{b}$

15. wio Group Title

$r(t) = \begin{bmatrix} 1 \\ 4 \\ -12 \end{bmatrix} t + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} t \\ 4t \\ -12t \end{bmatrix}$

16. theanonymous27 Group Title

Thank you!, One last question... Why did you pick x=1 first? what would have happened if you choose y=1 instead

17. wio Group Title

If you pick $$y=1$$ then you get $$x=1/4$$ and $$z=-3$$

18. wio Group Title

It's the same direction.

19. theanonymous27 Group Title

Thanks!