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Finding the point of intersection and distance of a vector

  1. Sep 18, 2011 #1
    1. The problem statement, all variables and given/known data
    Let P be the plane of Problem 9 and let L be the line perpendicular to P and passing through the point p=(1,1,2). Find the point of intersection between P and L and use it to compute the distance from p to P.

    Question 9 is this:
    Use the cross product to obtain the normal and standard equations of the plane P of problem 5.

    Question 5 is this:
    Verify that the points p_1=(0,0,1), p_2=(1,1,1) and p_3=(-1,0,1) are not aligned, and find the vector parametric equation of the plane P in R^3 that passes through them.


    2. Relevant equations
    Would I have to the the standard/normal equation from question 9 or the vector parametric equation from question 5?

    How do you integrate the point, p=(1,1,2) into the any of the above equations?

    Would the line perpendicular to P be similar to the normal found in question 9?

    3. The attempt at a solution
    For question 5:
    q=P_1+(P_2-P_1)
    q=(0,0,1)+t(1,1,0)
    (-1,0,1) = (0,0,1)+t(1,1,0)

    -1 = 0+1t --> -1=t
    0=0+1t --> 0=t
    1=1+0t --> 1=1

    Therefore they are not aligned.

    Vector parametric equation:
    [x,y,z]=q=P_1+t(P_2-P_1)+s(P_3-P_1)
    [x,y,z]=q=(0,0,1)+t(1,1,0)+s(-1,0,0)

    For question 9:
    P_1P_3 = (-1,0,0) P_1P_3 = (1,1,0)

    I used the cross product to find the normal to be (0,0,-1) then plugged it into the equation:
    a(x-x_0)+b(y-y_0)+c(z-z_0) = 0

    and then i got:

    0x+0y-z+1 = 0 as my standard equation

    I honestly have no idea where to begin for problem 10. I tried placing the point (1,1,2) into both equations, and I'm not sure if I am supposed to do that, and if I was, I wouldn't know how to use it to find the distance. :/
     
    Last edited: Sep 18, 2011
  2. jcsd
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