Writing an alternate vector Equation for a line.

[Physics345]

Homework Statement

Write an alternate vector equation for the following line. Change both the point and the direction vector:
w⃗ =(4,−1,3)+t(−2,1,7)

Homework Equations

The Attempt at a Solution

Did I write a proper alternate vector equation here? I'm still new to vectors in 3-space any tips or ways to confirm the answers to these types of questions would be greatly appreciated

 

[scottdave]

That's one way to do it. One way to check yourself is to find a new point on one of the lines (using another value of s, for example) then see if you can find a value for t which gets on that same point.

 

[Physics345]

basically a trial and error method correct for example:
=(-4,-1,3)+3(-2,1,7)
=(-4,-1,3)+(-6,3,21)
=(-10,2,24)

=(2,0,10)+3(-4,2,14)
=(2,0,10)+(-12,6,42)
and keep going till they match?
Am I on the right track here?

 

[Orodruin]

For a given t you should be able to write down an equation for s to get to the same point. You do not have to select a particular t, you can solve it generally.

This is also a faster way to solve the problem. Choose a new curve parameter $s$ such that $t = ks + m$ for some non-zero constants $k\neq 1$ and $m$.