10.7#16

Hi, Professor Taylor! I have done this problem several times, and I don't understand how to do it. I thought to write an equation for a line in 3D, you just needed two points for example: P: (a,b,c) Q: (d, e, f) r(t)= (a, b, c) + <d-a, e-b, f-c> t I am confused as to why the equation would be different for each different value of t given for the different points or vectors. Also, I know that this problem/homework set is due tonight, and it is pretty late, so I will get it wrong.

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Okey dokey....it looks like you have it right now, and we did talk about this in class: if you have a parameterization r(t)=v_0+tv_1 then you can replace v_0 by any other point on the line and replace v_1 by 𝝀v_1 for any nonzero  𝝀 and still have a parameterization.  So, you have this line in space, you should be able to imagine an ant walking along it, starting with various different times and places and going with different speeds. For instance to get from <2,-5,-5> at time 0 to <3,-5,-5> at time 7, the change in position is <3,-5,5> - <2,-5,5> = <1, 0, 0>, so it can get there by going at a velocity of <1/7mph, 0mph, 0mph> for seven hours.  Likewise we could go from <2,-5,5> at time 6 to <3,-5,5> at time 9 as if we're going between the same points in space from time 0 to time 3: r(t)=<2,-5,5>+t<1/3,0,0>. If you just replace t by t-6 this you use  r(t)=<2,-5,5>+(t-6)<1/3,0,0> to get the job done between t=6 and t=9.  You could also use the distributive law to write this as r(t)=<2,-5,5> -6<1/3,0,0> + t<1/3,0,0>= your part b).  Part c) just multiplies by 𝝀=-1, which is nonzero.