1. ## 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.
------------------------------------

2. 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 ...

3. Hello !

Originally Posted by o_O
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

4. Originally Posted by wardj
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.
------------------------------------
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!

5. But the OP asked for the area of $\triangle OAB$ ...

6. Originally Posted by o_O
But the OP asked for the area of $\triangle OAB$ ...

7. Originally Posted by o_O
But the OP asked for the area of $\triangle OAB$ ...
Where are my glasses?

8. 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?

9. Originally Posted by wardj
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... ~)

10. 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.

11. Your numbers should work then. I get around 28 as well.

12. 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.

13. 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!