## Area of Triangle with Cross Product: Equation Variations

Hello!

I'm trying to understand how this formula: 1/2 magnitude of v × w (area of triangle)
yields the same value no matter which 2 adjacent sides are chosen.

How would you prove mathematically that this is the case?
 PhysOrg.com science news on PhysOrg.com >> City-life changes blackbird personalities, study shows>> Origins of 'The Hoff' crab revealed (w/ Video)>> Older males make better fathers: Mature male beetles work harder, care less about female infidelity
 Blog Entries: 27 Recognitions: Gold Member Homework Help Science Advisor Hi Neen87! Welcome to PF! Hint: call the vertices a b and c, so the sides are a - b etc.

Recognitions:
Gold Member
Homework Help
 Quote by Neen87 Hello! I'm trying to understand how this formula: 1/2 magnitude of v × w (area of triangle) yields the same value no matter which 2 adjacent sides are chosen. How would you prove mathematically that this is the case?
Because if you draw the parallelogram with sides v and w, the cross product magnitude gives:

$$|v \times w| = |v||w|\sin\theta$$

where $\theta$ is the angle between the two vectors you have chosen for sides. Now, whichever two sides you choose and whichever direction they point, the angle between them will be either $\theta$ or $\pi - \theta$. Either way you get the same value for its sine.

## Area of Triangle with Cross Product: Equation Variations

Thanks so much! :-)

 Tags area of triangle, cross product, proofs

 Similar discussions for: Area of Triangle with Cross Product: Equation Variations Thread Forum Replies Introductory Physics Homework 1 Calculus 1 Advanced Physics Homework 7 Calculus & Beyond Homework 4 Calculus & Beyond Homework 0