MHB How to determine a vector parallel to another vector.

[Drain Brain]

I wonder how my book did the calculation for this example

$\vec{r}=<5,2-t,10+4t>$ the vector parallel to it is $\vec{a}=<0,-1,4>$

can you show the workings of this example! TIA!

 

[Prove It]

I wonder how my book did the calculation for this example

$\vec{r}=<5,2-t,10+4t>$ the vector parallel to it is $\vec{a}=<0,-1,4>$

can you show the workings of this example! TIA!

$\displaystyle \begin{align*} \mathbf{r} &= \left( 5, 2 - t , 10 + 4t \right) \\ &= \left( 0 , -1, 4 \right) \, t + \left( 5 , 2, 10 \right) \end{align*}$

It is clearly parallel to $\displaystyle \begin{align*} \left( 0, -1, 4 \right) \end{align*}$, just scaled up by some factor t, and moved 5 units in the x direction, 2 units in the y direction, and 10 units in the z direction.