I was helping a friend in grade 12 with his Linear Algebra, when I got stumped on some questions I wasn't quite to sure of. I forgot some of my old linear algebra so I need you guys to help me out:) Here's the question:
1)The two line with equations L1: r = (1,2,-4) + t(k+1,3k+1,k-3) and L2: x=2-3s, y=1-10s, z=3-5s are given.
a) Determine the value of k if these two lines are parallel and perpendicular.
a)does the point P(2,4,2) lie on the line x=2t,y=3+t, z=1+t?
b)if the point (a,b,-3) lies on the line find values of a and b
3) Determine the equation of the plane through A(2,1-5) perpendicular to both 3x-2y+2=8 and 4x+6y-5z=10.
4) What is the scalar equation of the plane parallel to the plane 3x-9y+z-12=0 and including the point (-3,7,1).
2. The attempt at a solution
1. Parallel : I took the t(k+1,3k+1,k-3) as tv(vector) and since it had to be parallel with the line, I took the vector from the other equation x=2-3s, y=1-10s, z=3-5s, which is -3,-10,-5 and i equated to the respective x y z in the first line. So:
k+1 = -3 -> -4
3k+1 = -10 -> -11/3
k-3 = -5 -> -2
Perpendicular: I need your help I didn't know how to do it, do I have to make it somehow equal 0? Something about dot product being 0 not sure need advice guys!
a)So i subed in 2, 4 ,2 into the respective x y and z, if they all had the same t I said the point lied on the same line, and it did.
b) I subed in -3 for z = 1+t , solved for t, then used that t to solve for the x and y in the other provided equations.
3. I took the vectors from the plane equations so v1 = (3,-2,2) and v2= (4,-6,-5) then I did cross product and I got v3 = ( -2,23,-10)( someone confirm this). After that I made it into the equation -2+23y-10+D = 0, I put in the points it went through into their respective x y and z coordinated and solved for z.
4) A bit confused on this one can someone give me a tip I think it's just basically the same equation except you'd have to solve for a new D, so 3x-9y+z-12=0 would be 3x-9y+z-D=0 then you sub in the new points solve for a new D. I think that's how you do it. Can you guys help me:)
Thanks a lot guys I appreciate every response in advance.