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Using cross product to find angle between two vectors

  1. Jun 30, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the angle between
    [tex]\begin{align*}
    \vec{A} = 10\hat{y} + 2\hat{z} \\
    and \\
    \vec{B} = -4\hat{y}+0.5\hat{z}
    \end{align*}[/tex]
    using the cross product.

    The answer is given to be 161.5 degrees.

    2. Relevant equations
    [tex]
    \left| \vec{A} \times \vec{B} \right| = \left| \vec{A} \right| \left| \vec{B} \right|sin(\theta)

    [/tex]

    3. The attempt at a solution
    [tex]
    \left| \vec{A} \times \vec{B} \right| = [/tex] [tex]\left|
    \begin{array}{ccc}
    \hat{x} & \hat{y} & \hat{z} \\
    0 & 10 & 2 \\
    0 & -4 & 0.5
    \end{array} \right| = \left| 13\hat{x} \right| = 13 [/tex]

    The magnitude of A cross B is 13.

    Next we find the magnitude of vectors A and B:
    [tex] \left| \vec{A} \right| = \sqrt{10^2+2^2} = \sqrt{104} = 10.198039 [/tex]
    and
    [tex] \left| \vec{B} \right| = \sqrt{(-4)^2+(\frac{1}{2})^2} = \sqrt{16.25} = 4.0311289 [/tex]

    multiplying the previous two answers we get:
    41.109609

    So now we should have:
    [tex] \frac{13}{41.109609} = sin(\theta) [/tex]

    Solving for theta, we get:
    18.434951 degrees.


    This is frustrating: 180-18.434951 = the correct answer. I'm not quite sure where I'm going wrong here.

    I must be making the same mistake repeatedly. Another problem was the same thing, but with the numbers changed, and I also got the 180-{the answer I was getting} = {the correct answer}, but when I tried the example using the SAME methodology, I got the correct answer.

    Can someone please share some relevant wisdom in my direction?
     
  2. jcsd
  3. Jun 30, 2011 #2

    ehild

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    Homework Helper
    Gold Member

    sin(alpha)=sin(180-alpha) Plot the two vectors and you will see what angle they enclose.


    ehild
     
  4. Jun 30, 2011 #3

    I like Serena

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    Homework Helper

    You might use the sign of the inner dot product to see which angle you have.
     
  5. Jun 30, 2011 #4
    I can plot them, and I can see the angle, but I'm interested in calculating the angle.
    When I use the dot product I get the correct result, but I cannot see where my mistake is while using the cross product.
     
  6. Jun 30, 2011 #5

    ehild

    User Avatar
    Homework Helper
    Gold Member

    There is no mistake, you get the sine of the angle, but there are two angles between 0 and pi with the same sine.

    ehild
     
  7. Jul 1, 2011 #6
    Oh wow; I didn't even consider that the answer wasn't unique.
    Thanks!
     
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