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12) Find the parametric equations for the line through the point P = (6, 1,2) that is perpendicular to the plane &x + 3y + 3z = 1....

Question

12) Find the parametric equations for the line through the point P = (6, 1,2) that is perpendicular to the plane &x + 3y + 3z = 1.

12) Find the parametric equations for the line through the point P = (6, 1,2) that is perpendicular to the plane &x + 3y + 3z = 1.



Answers

Find parametric equations for the line through the point $(0,1,2)$ that is parallel to the plane $x+y+z=2$ and perpendicular to the line $x=1+t, y=1-t, z=2 t$

We are given descriptions of a line and rest to find Parametric equations for this line. So in part they were told to the line. L goes through points P with coordinates 132 and cue with coordinates to five negative six. First, we'll find a direction. Vector he just the directive Vector from P to Q. Find this by tracking the initial point from the terminal Point. Let me get the vector of two minus one, which is one five minus three or two and negative six minus two or negative eight. Rich in standard form. This is I plus two J minus eight K, and therefore we have that the align through these two points lft. This is equal to the vector t plus we'll use point p one to t plus three and negative 80 plus two. Or, if you wanted to write it in standard form, this is equal to t plus one I plus two t plus three Jay plus Negative 80 plus two. Okay, then in part B, we're asked to find the equation of the line containing the point P with coordinates one negative 24 and also perpendicular to the plane H where h is given by the equation three x plus five y plus seven z equals 15 Mhm. Now, because the line l is perpendicular to H, it follows the line. L is in the same direction as the normal vector. So l is in same direction as the normal vector to H, which is the Vector three five seven. And therefore, this is a direction vector for our line and we have a point on the line. So there are lined L f t is equal to the vector three t plus one, five T minus two and 70 plus four or in standard form, this is three T plus one. Hi plus five T minus two Jay plus 70 plus four. Okay.

Here in this problem, we have given the line passes through the point so point is given edge minus 6 to 3 and parallel to the line. So line equation is given edge. One by two works is equal to one by three. Y is equal to that plus one so it can be rewritten. Edge X Y two is equal to why by three is equal to zero plus one by one. Now going forward, I can write the value as here the value. It's not difficult to minus six. Why not? Difficult to They are not difficult to three, so I can write. Accessible to minus six plus two. T Why is equal to two plus three. T and D are difficult to three, plus the going forward and using this symmetric equation of the line. So the formula is given by X minus X not by a is equal to why minus why not by B is equal to their minus. They're not by sea, So just putting the value here So the question becomes X plus six by two is equal to y minus two by three is equal to zero minus three by one. So this is the symmetric key question of the given line

So now what we want to do is you want to find the line that passes to the .12 or six. And the spot particular to the line X. That's too white. Plus G equals 2 5. So it's a bit different from what we have done before but it's the same thing. So this is again my this guy and my vector B is just the coefficients of X. Y. And G. So this is one three and one. So the victory question is A. Plus tv. This is 106 nasty Times 1, 3 and one. There's a little equation if you write it in terms of high. So this is I. Plus judo Jessica less six K. Cup. Last two times I cap. That's T. Jacob last cake up. And then for the parametric equation. Yeah With a parametric equation. This is X zero over one. Is why my sorry X -1 Over one equal to 1 -0/3 is equal to G -6/1. And then you get it to A. T. I should do that. Then we just solve it. I was just write it and blow here on the side. So excess D. Plus like one. Why is three D. And G. S. T. Plus six. This is a victory question. That's it.

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


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