What is the equation for the area of a triangle in terms of its sides and angle?

[ozon]

Hello,
I am new at this forum. Firstly, I want to say I am happy to be a member of this forum. English is not my native language. So I can make some grammar and other mistakes. I try to write with no mistakes. If I make mistakes for writing, I apologize.
I have a question. You see a triangle in the picture. There are some vectors. h is height or short distance between p₀ and V₃. In the picture, you see an equation to find h. How do you prove this equation?

 

[Simon Bridge]

Welcome to PF;
Have you tried starting with the definition of a vector cross-product?

 

[ozon]

Thanks. Yes, I have. But I can't solve it.
h is norm of short distance between p₀ and V₃. So I have to use a new symbol for short distance. I want to call it as V₄.
h=|V₄|
|V₄||V₃|sin90⁰=|V₁xV₂|
V₄xV₃=|V₁xV₂|
I have tried something but I have no idea. Some clues are very useful to prove this equation.

 

[Fightfish]

HINT:
What does $h |V_{3}|$ give you?
What does $|V_{1} \times V_{2}|$ give you?
It is related to a property of the triangle.

 

[ozon]

HINT:
What does $h |V_{3}|$ give you?
What does $|V_{1} \times V_{2}|$ give you?
It is related to a property of the triangle.

Thanks but I do not have any idea.

 

[Fightfish]

Okay, I'll be a little more explicit: how would you go about finding the area of the triangle? There are two ways: 1st is by geometrical means, and 2nd by vectorial methods.

 

[ozon]

Ok. First way is not problem to find area of the triangle. But I do not know how to find area of the triangle with vectorial methods. I think second way is necessary to prove this equation. Thanks for your clues. But I need more than them.

 

[Chestermiller]

Ok. First way is not problem to find area of the triangle. But I do not know how to find area of the triangle with vectorial methods. I think second way is necessary to prove this equation. Thanks for your clues. But I need more than them.

If A and B are the lengths of two sides of a triangle, and θ is the angle between the two sides, what is the equation for the area of the triangle in terms of A, B, and θ?

 

[ozon]

If A and B are the lengths of two sides of a triangle, and θ is the angle between the two sides, what is the equation for the area of the triangle in terms of A, B, and θ?

Your clue is very helpful for me. Thanks. It is very easy now.
Area of triangle (1/2)h|V₃|sin90⁰=(1/2)|V₁||V₂|sinθ
h|V₃|=|V₁xV₂|
h=(|V₁xV₂|)/|V₃|
Thank you everybody for your helping.