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Homework Help: Vectors help

  1. Apr 30, 2010 #1
    I am working on 2 problems and wanted to know your thoughts on them:

    Problem 1
    Given the vectors x = <3,2,-4>, y = <-3/2,1,-2>, and z = <0,2,1>, select all statements below that apply.
    A.
    the vectors x and y are orthogonal.
    B.
    the vectors x and y are in opposite directions.
    C.
    the vectors x and z are orthogonal
    D.
    the vectors x and z are in opposite directions


    2 Vectors are orthogonal if their dot product is 0.
    2 vectors are orthogonal is the angle between them is 180.

    To determine the angle between 2 vectors we can use the equation Cos ⊖= The dot product of the 2 vectors/ Distance of vector1 * distance of vector 2


    A.
    the vectors x and y are orthogonal.
    The dot product of x and y is ( 3* -3/2 , 2*1, -4*-2) = (-9/2 + 2 + 8) = 11/2. hence x and y are not orthogonal

    B.
    the vectors x and y are in opposite directions.

    Cos ⊖= 11/2 / (sqrt ( 29)* (sqrt(29)/ 2)) = 11/2 / 29/2 = 11/29
    Cos ⊖ = 11/29 = approx 67.71 degrees. x and y are not in opposite directions.

    C.
    the vectors x and z are orthogonal
    The dot product of x and z is ( 3* 0 , 2*2 , -4*-1) = (0 + 4 - 4). x and z are orthogonal.


    D.
    the vectors x and z are in opposite directions

    Cos ⊖= 0 = 90 degrees. This confirms our answer in part C that x and z are orthogonal and create a right angle. they are not in opposite directions.



    Problem 2
    Given the vector x = <3,2,-4>,

    1. find the length of the vector x (write sqrt(#) for square root to write your answer exactly).

    2. find the unit vector in the same direction as x. (Write your answer exactly using the form <v1,v2,v3,...> , rationalize all denominators)

    1) The length of a vector is the square root of the sum of squares. The length is sqrt ( 3^2 + 2^2 +(-4)^2 = sqrt ( 29)

    2) The unit vector in the same direction of x is given by x/ length of x. We have ( 3, 2, -4) / sqrt(29) from part A. So we have ( 3*sqrt(29)/ 29 , 2*sqrt(29), -4*sqrt(29)/ 29)
     
  2. jcsd
  3. Apr 30, 2010 #2
    Your calculations look fine to me. I didn't double-check anything with a calculator, but the answers I found agree with yours.

    On your last problem, your unit vector is incorrect, possibly you forgot to type in a number.

    Also, this statement is incorrect: "2 vectors are orthogonal is the angle between them is 180."

    And, when calculating the dot product of two vectors and writing the multiplication of each term, I would be careful not to write it as a vector.

    Example: It is not really a good habit to write: (a,b,c) · (x,y,z) = (ax,by,cz) = ax + by + cz. It can be simply written: (a,b,c) · (x,y,z) = ax + by + cz.
     
    Last edited: Apr 30, 2010
  4. Apr 30, 2010 #3
    Thanks

    did you mean ( 3*sqrt(29)/ 29 , 2*sqrt(29), -4*sqrt(29)/ 29) is incorrect? I will re check that problem if so
     
  5. Apr 30, 2010 #4
    Yes, it is, due to a minor error.
     
  6. Apr 30, 2010 #5
    youre right
    it should be ( 3*sqrt(29)/ 29 , 2*sqrt(29)/ 29, -4*sqrt(29)/ 29)
     
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