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Angle Between 2 Vectors

  1. Feb 16, 2009 #1
    1. Find the SMALLEST angle between the vectors T and S

    Given vectors T = 2ax — 6ay + 3az and S =ax + 2ay + az,

    See the thing im confused about is whether to use Cross Product or Dot Product. I used the dot product formula

    TdotS = |T||S|cos

    and solved for cos theta ((theta = cos-1))

    I got 114 degrees

    The solution I have uses CROSS PRODUCT and finds an angle 65 Degrees

    I dont get why the cross product would give a smaller angle? Can anyone tell me

    If i take 114 - 180 i get -66 but I dont get why I would subtract 180 *and also its a negative angle then..HELP!
    Last edited: Feb 16, 2009
  2. jcsd
  3. Feb 16, 2009 #2


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    Given vectors T = 2ax — 6ay + 3az and 8 = 3^-4- 2ay + az,

  4. Feb 16, 2009 #3
    sorry about that, fixed now..
  5. Feb 16, 2009 #4


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    Can you show us how you did?
  6. Feb 16, 2009 #5
    ok sure its a simple dot product tahts why i didnt show it my main question is can we go from an angle of 114 to 66..and if its because of 180-114 what would be the reason for subtracting 180?

    Anyways ill show it

    Given vectors T = 2ax — 6ay + 3az and S =ax - 2ay + az,

    T dot S = 2 -12 + 3 = -7

    |T| = sqrt 49 = 7
    |S| = sqrt 6

    cos -1(-7 / (7 * sqrt(6) ) = 114

    the solution I have uses the cross product and the angle they get is 66 degrees
    Last edited: Feb 16, 2009
  7. Feb 16, 2009 #6


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    if you draw the situation on a paper, you will loose the confusion.

    T dot S = 2 + 12 + 3 = 15 (IT IS 17)
  8. Feb 16, 2009 #7
    im sorry i drew it out but i dont see how this works....

    when i take the dot product and cos inverse i get 114 so this is not the angle between the two vectors?

    http://img18.imageshack.us/img18/2420/95148869qw3.jpg [Broken]

    obviously from the picture it seems the angle is infact 66 degrees but why then mathematically i get 114 degrees using the dot product?

    I thought that angle that i get from the dot product is the angle between the vectors so why did I get 114...:(
    Last edited by a moderator: May 4, 2017
  9. Feb 16, 2009 #8


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    if the angle is 114 also 180 -114 = 66 degree is the angle between the vectors.

    Now I get with my calculator that [itex]\text{arccos} (17/(7\sqrt{6})) = 7.5[/itex] degree.

    Can you outline the solution using cross product for me?
  10. Feb 16, 2009 #9
    yea sorry i put int he wrong component..its

    S =ax + 2ay + az,
  11. Feb 16, 2009 #10
    ^now the new picture looks like the angle IS 114 but i dont see how the `smallest`is 66 degrees...

    I do see that the angle 66 degrees is made with the vector S and NEGATIVE T

    but S and T is 114...the cross product method is just the general cross product and then take sin -1

    I just use my CASIO calculator it does the cross product

    THOUGH YOU TAKE MAGNTIUDE OF (T X S) = |T||S| sin angle

    sin -1 will give u 66 degrees!
  12. Feb 16, 2009 #11
    Remember, 114 has the same sin as 66 degrees! Cos is maybe easier to use for this problem as it is uniquely valued in the 0°-180° region. Whenever you use trig formulas, make a habit of remember that ALL the trig formulas are multivalued!
  13. Feb 16, 2009 #12
    And you're drawing is incorrect.

    edit: The 3d plot shown above IS correct, sorry.
  14. Feb 16, 2009 #13
    but cos of (66 ) = 0.4

    cos of (114 ) = -0.4

    they are not the same
  15. Feb 16, 2009 #14
    That is correct, they are not, that is why the dot product gives you the correct answer.

  16. Feb 16, 2009 #15
    Remember that cosine takes values of 1 to -1 in the 0° to 180° range, but sin is double valued in that region, that is: it takes on each value between 0 and 1 twice. So if you want to use sine, you have to ask yourself at the end of the problem if it could possibly be 114° instead of 66°. If you use cosine, you know you have the right answer. You could check by taking the dot product of the two vectors, you will find that it is negative and hence [tex]\theta > 90[/tex]°
  17. Feb 16, 2009 #16
    im confused still you said the dot product is the correct answer? So on a test if it said find the angle between two vectors..

    from the cross product the angle is found to be 66

    from the dot product the angle is found to be 114

    What is the right answer?

  18. Feb 16, 2009 #17


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    two vectors have two angles, which sum is 180 degrees.

    Draw two lines (2D vectors) in the plane.
  19. Feb 16, 2009 #18
    I have always been asked to find the angle between the two vectors if they are placed tail to tail, which the dot product gives the correct answer for and the cross product does too, but your CALCULATOR gives you the wrong answer for the sin. Use the dot product, this is a classical application of the dot product.
  20. Feb 16, 2009 #19
    yes that``s what i was wondering isn``t the angle used in the dot and cross product the angle between the vectors placed TAIL TO TAIL....

    if that is true then the angle is 114.
  21. Feb 16, 2009 #20
    yes! That's exactly right, the vectors need to be placed tail to tail in all questions I have ever done.

    Make sure you understand why the sine didn't give you the correct answer, its only because each angle in the 0° - 180° range is duplicated. Lets say I tell you that the sine of the angle between two vectors is 0.9. Well then you should punch it into your calculator and see that the 'angle' is 64.2° using inverse sine, but I could say it could also be 115.8, it has the same sine! Cos on the other hand only has one answer for these problems, thats why you should use the dot product.
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