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Math Help - Vectors question

  1. #1
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    Vectors question

    Hello,
    I can't get the correct answer for this and was wondering if anybody could help.
    ------------------------------------
    OA = 5i + 6j - 3k
    OB = 3j + 6k
    OC = 45i - 30j + 15k

    Calculate the area of triangle OAB.
    ------------------------------------
    Thanks for any answers!
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  2. #2
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    Have you learned the cross product yet?

    The area of a triangle is given as: \frac{1}{2} \: \big| \big| u \times v \big| \big|

    where u and v are vectors whose inital points coincide.

    Here, you take \vec{OA} \times \vec{OB} = (5,6,-3) \times (0,3,6) = \left(\left| \begin{array}{cc} 6 & -3 \\ 3 & 6 \end{array}\right|, \left| \begin{array}{cc} 5 & -3 \\ 0 & 6\end{array}\right|, \left| \begin{array}{cc} 5 & 6 \\ 0 & 3\end{array}\right|\right) .

    Take the half the norm and youre' done.

    Don't think OC was needed here ...
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  3. #3
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    Hello !

    Quote Originally Posted by o_O View Post
    Have you learned the cross product yet?

    The area of a triangle is given as: \frac{1}{2} \: \big| \big| u \times v \big| \big|

    where u and v are vectors whose inital points coincide.

    Here, you take \vec{OA} \times \vec{OB} = (5,6,-3) \times (0,3,6) = \left(\left| \begin{array}{cc} 6 & -3 \\ 3 & 6 \end{array}\right|, \left| \begin{array}{cc} 5 & -3 \\ 0 & 6\end{array}\right|, \left| \begin{array}{cc} 5 & 6 \\ 0 & 3\end{array}\right|\right) .

    Take the half the norm and youre' done.

    Don't think OC was needed here ...
    The area of a triangle ABC is given by \frac 12 ||\vec{AB} \times \vec{AC}||

    not by the cross product of the vectors from the origin... it would be nonsense



    You can use this formula : Heron's formula - Wikipedia, the free encyclopedia, but I doubt it's what you're being expected
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  4. #4
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    Quote Originally Posted by wardj View Post
    Hello,
    I can't get the correct answer for this and was wondering if anybody could help.
    ------------------------------------
    OA = 5i + 6j - 3k
    OB = 3j + 6k
    OC = 45i - 30j + 15k

    Calculate the area of triangle OAB.
    ------------------------------------
    Thanks for any answers!
    The area of the triangle is calculated by:

    a_{ABC}=\frac12 \cdot \left| \overrightarrow{AB} \times \overrightarrow{AC} \right|

    with

    \overrightarrow{AB}=\overrightarrow{OB} - \overrightarrow{OA} and

    \overrightarrow{AC}=\overrightarrow{OC} - \overrightarrow{OA}

    My final result is a_{ABC} = 15 \cdot \sqrt{406} \approx 302.242

    Too fast for me, Moo!
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  5. #5
    o_O
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    But the OP asked for the area of \triangle OAB ...
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  6. #6
    Moo
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    Quote Originally Posted by o_O View Post
    But the OP asked for the area of \triangle OAB ...
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  7. #7
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    Quote Originally Posted by o_O View Post
    But the OP asked for the area of \triangle OAB ...
    Where are my glasses?
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  8. #8
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    Yeah it's triangle OAB. OC wasn't needed for this part of the question (the question has several parts and I included it without thinking). This is the only one I haven't been able to get the right answer for. I tried to find the magnitude of the 3 sides, OA, OB and AB, which I made to be 70^.5, 115^.5 and 45^.5. I put those into the formula for the area of a triangle and came out with about 28, but the answer book says 188.25. So the question is what stupid thing have I done wrong?
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  9. #9
    Moo
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    Quote Originally Posted by wardj View Post
    Yeah it's triangle OAB. OC wasn't needed for this part of the question (the question has several parts and I included it without thinking). This is the only one I haven't been able to get the right answer for. I tried to find the magnitude of the 3 sides, OA, OB and AB, which I made to be 70^.5, 115^.5 and 45^.5. I put those into the formula for the area of a triangle and came out with about 28, but the answer book says 188.25. So the question is what stupid thing have I done wrong?
    Hmmmm the question is : what formula have you used ?

    Because you can't get an immediate formula of the area containing only the magnitudes of the 3 sides, unless this triangle is special

    (or it may be a complicated one... ~)
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  10. #10
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    According to my formula book area of a triangle = (s(s-a)(s-b)(s-c))^0.5

    Where s = (a + b + c)/2

    I thought a, b and c were the lengths of the sides of the triangle, which I assumed would be the magnitude of the vectors between the points.
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  11. #11
    o_O
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    Your numbers should work then. I get around 28 as well.
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  12. #12
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    Note: The first complete answer given by o_0 is the correct way to work this question.
    Moreover, the area of triangle OCB is 188.25.
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  13. #13
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    Ah ha. Thank you! I wasn't actually wrong (although I wasn't using the best method!). The question/answer was incorrect in my book. It asks for the area of triangle OAB, yet the answer given is the area of triangle OCB. I hadn't thought to work that area out, but you're right.

    Thanks for the help guys, seems like the book was wrong!
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