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Homework Help: 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|

    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

    b) Find a vector equation of the line through A and B
    my attempt:
    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


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    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


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