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Help with linear questions. Got wrong on test and need help correcting them

  1. Jul 22, 2012 #1
    Here are some questions that I received on my test. I got most of these wrong but I got a couple points here and there. Here are the questions I had trouble with. At the bottom on the questions I tried to solve them but I'm not sure if i did them correctly. Please try and help me because I have a final coming up this week and I want to learn how to solve these questions.

    1. Consider the three vectors u=(4,-1,-5) , v=(1,-4,1) and w= (1,1,-2)
    a) compute the scalar triple product of u,v,w.
    my attempt:
    u * (v X w)
    4 -1 -5
    1 -4 1
    1 1 -2

    =4(7) + 1(-3) -5(5)
    = |0|
    =0

    b) What can you deduce about the vectors u,v,w, supposing they have the same initial point????
    my attempt: That the answer would be different because it would be multiplying with different multiples…?

    2. Considers the points A(1,0,1) B(1,2,3) C(-1,0,2)
    a) find the angle between the vectors AB and AC
    my attempt:
    cos=AB*AC/||AB|| ||AC||
    =2/(8)^1/2 (5)^1/2
    = 0.3162
    =cos-1(0.3162)
    =71.56

    b) Find a vector equation of the line through A and B
    my attempt:
    A=(1,0,1)
    B= (1,2,3
    ????

    3. Considers the L1(line 1) with symmetric equation (x-1)/-1 = (y+2)/2 =z/3
    and the L2(line 2) parallel to v2=(1,0,-1) and throughout the point P2(0,1,0)

    a) Find the direction vector v1 for the line L1 and give a point P1 on L1
    my attempt:
    p1 (1,1, 1/3)
    v1 (1,1,1)

    b) Find a parametric equation of the line L2
    my attempt:
    v2= (1,0,-1)
    P2= (0,1,0)
    L2 should equal {t, 1, -1}

    c) show that the lines L1 and L2 are skew lines
    my attempt:
    |x1x2 * (V1 X V2)| \ ||V1 X V2||

    =(0,1,0) * ( -1, -2, -1) / 6^1/2
    = -2/6^1/2


    d) Find a unit vector u orthogonal to both v1 and v2
    my attempt:
    v1 (1,1,1)
    v2 (1,0, -1)
    v3 ( , , )
    v1 X v2
    1 1 1
    1 0 -1
    u=( -1,0,-1)

    e) find the orthogonal projection P1P2 on u
    (p1p2 * u/ ||u|| ) u
    =(-2/2^1/2, 0, -2/2^1/2)

    f) deduce the distance d=||Proju P1P || between the lines L1 and L2


    4. If u and v are vector in n-space, Simplify: ( u+v) * (u-v)
    my attempt:
    =uu –uv + uv - vv
    =||u||^2 - ||v||^2

    b) use your previous result to show that the parallelogram defined by u and v is a rhombus if and only if its diagonals are perpendicular.
    A rhombus has 4 sides that are equal and this would prove it.
     
    Last edited: Jul 22, 2012
  2. jcsd
  3. Jul 22, 2012 #2

    eumyang

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    Homework Helper

    Geometrically, the scalar triple product is the (signed) volume of the parallelepiped defined by the three given vectors. If the volume of this parallelepiped is zero, then...?
     
  4. Jul 22, 2012 #3
    it means that the parallelepiped is planar and has no volume. This means that the given three vectors are linearly dependent???
     
  5. Jul 22, 2012 #4

    eumyang

    User Avatar
    Homework Helper

    Yes.
     
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