document.write( "Question 1121153: I'm working on some summer problems so that I can be more prepared when I go into my class in the fall. (I haven’t learned any of this, but I am trying to become familiar with it.) I found a website full of problems of the content we will be learning, but it doesn't have the answers. I need a little guidance on how to do this problem. Here is the problem:\r
\n" ); document.write( "\n" ); document.write( "I’m going to do my best to talk through each part and tell you how I think it should be solved!\r
\n" ); document.write( "\n" ); document.write( "A line L1 passes through points P(−1,6,−1) and Q(0,4,1).
\n" ); document.write( "a. (i) Show that vector PQ=(1,−2,2).
\n" ); document.write( "So, I can see that if you take the x, y, and z of Q, and subtract the x, y and z of P, then you get PQ. But, is there some sort of formula for this that you would use to answer this question?
\n" ); document.write( "(ii) Hence, write down an equation for L1 in the form r=a+tb.
\n" ); document.write( "Would this be: r=(−1,6,−1)+t(0,4,1)? (I just plugged in P for a and Q for b.
\n" ); document.write( "b. A second line L2 has equation r=(4,2,−1)+s(3,0,−4).
\n" ); document.write( "Find the cosine between vector PQ and L2.
\n" ); document.write( "I’m super confused on how you could find the cosine of this when all you have is some points? Or, if I draw this out somehow would I find the sides to do the Law of Cosines?\r
\n" ); document.write( "\n" ); document.write( "c. The lines L1 and L2 intersect at the point R. Find the coordinates of R.
\n" ); document.write( "Would I use a midpoint formula to discover this? Since R would be in the middle of both lines? \n" ); document.write( "
Algebra.Com's Answer #736973 by math_helper(2461)\"\" \"About 
You can put this solution on YOUR website!
a.(i) You are correct. The formula, given points P(x1,y1,z1) and Q (x2,y2,z2) is v = < x2-x1, y2-y1, z2-z1 >. for a vector that points in the direction from P toward Q.
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\n" ); document.write( "\n" ); document.write( "a. (ii) They want a parametric representation of the line. What you wrote is essentially correct, however, be aware that you are specifying vectors, not just coordinates. Sure, once you evaluate a vector equation to a resultant < a,b,c > you can say the tip of the resultant vector is at the point (a,b,c), but be sure to keep vectors straight. Here you can take one point as the start \"+r%5B0%5D+\" (say P) and then use the vector you found in (a (i)) multiplied by a parameter t: r = < -1, 6, -1 > + t* < 1,-2,2 >, so ( here, -1+t is the x component, 6-2t is the y component, and -1+2t is the z component ). t varies from \"-infinity\" to \"infinity\"
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\n" ); document.write( "\n" ); document.write( "b. The 2nd line has form: \"++r%5B2%5D+\" = <4+3s, 2, -1-4s>\r
\n" ); document.write( "\n" ); document.write( "The \"dot\" product of two vectors is A ⋅ B = where \"theta\" is the angle between the two vectors, \"+abs%28A%29+=+sqrt%28A%5Bx%5D%5E2+%2B+A%5By%5D%5E2+%2B+A%5Bz%5D%5E2%29+\" and \"+abs%28B%29+=+sqrt%28B%5Bx%5D%5E2+%2B+B%5By%5D%5E2+%2B+B%5Bz%5D%5E2%29+\", gives a SCALAR measure of how much vectors A and B point in the same direction. So you have all the numbers needed. Let A be the vector PQ you found, and let B be the vector <3,0,-4> (the part that 's' scales). This is so because s<3,0,-4> and t<1,-2,2> have the same angle between them as the two lines PQ and L2.
\n" ); document.write( " ( If this last fact is not clear, watch this video https://www.youtube.com/watch?v=PyPp4QvQY3Q ) \r
\n" ); document.write( "\n" ); document.write( "You will compute the numbers and solve for \"+cos%28theta%29+\"
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\n" ); document.write( "\n" ); document.write( "c. At the point of intersection, \"+r%5Bx%5D+=+r%5B2x%5D+\", \"r%5By%5D+=+r%5B2y%5D+\" and \"+r%5Bz%5D+=+r%5B2z%5D+\" (where \"r%5B2x%5D\" is supposed to be r sub 2, sub x, etc.). Each of the lines (a(ii)) and (b) can be thought of as the tip of a vector traced out as t (s) varies.
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\n" ); document.write( "\n" ); document.write( "Setting x components equal: -1+t = 4+3s
\n" ); document.write( "Setting y components equal: 6-2t = 2
\n" ); document.write( "Setting z components equal: -1+2t = -1-4s
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\n" ); document.write( "\n" ); document.write( "The y component equation gives t=2, substituting into the x component equation
\n" ); document.write( "gives s=-1 (and these values work in the z component equation, so it is a good solution).
\n" ); document.write( "To find the coordinates, substitute t=2 in part a(ii) and it should match the coordinates when
\n" ); document.write( "you set s=-1 in part b. You should find they intersect at (1,2,3).
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\n" ); document.write( "\n" ); document.write( "Hopefully Theo will agree with these solutions. Maybe he can explain it better or make any needed corrections. I was rusty on it as well. \r
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