Using cross product to find angle between two vectors

  1. 1. The problem statement, all variables and given/known data
    Find the angle between
    \vec{A} = 10\hat{y} + 2\hat{z} \\
    and \\
    \vec{B} = -4\hat{y}+0.5\hat{z}
    using the cross product.

    The answer is given to be 161.5 degrees.

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


    3. The attempt at a solution
    \left| \vec{A} \times \vec{B} \right| = [/tex] [tex]\left|
    \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]
    [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:

    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. ehild

    ehild 12,698
    Homework Helper
    Gold Member
    2014 Award

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

  4. I like Serena

    I like Serena 6,181
    Homework Helper

    You might use the sign of the inner dot product to see which angle you have.
  5. 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. ehild

    ehild 12,698
    Homework Helper
    Gold Member
    2014 Award

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

  7. Oh wow; I didn't even consider that the answer wasn't unique.
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