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Problem 16 Medium Difficulty

(a) Find parametric equations for the line through $ (2, 4, 6) $ that is perpendicular to the plane $ x - y + 3z = 7 $.
(b) In what points does this line intersect the coordinate planes?

Answer

(A). $\begin{aligned} x &=2+t \\ y &=4-t \\ z &=6+3 t \end{aligned}$
(B). In the $x y$ and $y z$ -planes: $(0,6,0)$
In the $x z$ -plane: $(6,0,18)$

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Video Transcript

Okay. So what we want to do is you want to want to find the the equation of the line that passes to 246 and is perpendicular to x minus Y plus three. G equals for seven So fast. We need to find the normal veteran which is 1 -1 and three. So then the parametric equation is given by a plus tion. Which is to wake up plus forge a couple six K cup plus two times. Take up by the state couples take a gap So the parliament take your question then becomes X equals to two Christie. Why he calls to form honesty and Dick was just X. Plastic. That answers the first part that planted the second part. We need to find where this line intersects the planes. So first we find where it intersects the excitement. If you wanted to intercept the X ray plain we must set due to be zero. So this means that 30-plus 6 must be zero. That gives us that team must be misunderstood. So if these miners to what happens to access to plus city which is two plus minus or zero, what happens to why? Why is four minus T. Four minus of minus 24 plus 26. What happens to G. Six plus three D. But we set it to zero already. So it is zero. So intersects the X. Y. plane at the .06 and zero. Then you go and see where it intersects the Y. Z. Plan again. To intercept the Y. Z. Plan X must be zero. So two plus two must be zero. So that gives us the team must be minus two. Now if X0 why is four minus 15 is minus two then this is four minus minus two is six. G six plus 36 plus two times minus two is six minus +60. So interested the YG plan at the .06 and zero. Similarly, if we wanted to intercept the exit plan you must set right to be zero. Which means that four -T must be zero. That tells us that he must be four. So what happens to X 50 is for access to two plus 32 plus 46. Why we search? It'll be zero already. That is that there's no worrying about it and then G six plus 30 which is six plus four times 36 plus 12 18. So intersex txt plan at 601 18 and that's it. That's the