Area of Triangle with Cross Product: Equation Variations

In summary, the formula 1/2 magnitude of v × w (area of triangle) yields the same value regardless of which two adjacent sides are chosen. This can be proven mathematically by considering the parallelogram formed by these two sides, where the cross product magnitude will always result in the same value due to the angle between the two sides.
  • #1
Neen87
8
0
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?
 
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  • #2
Welcome to PF!

Hi Neen87! Welcome to PF! :smile:

Hint: call the vertices a b and c, so the sides are a - b etc. :wink:
 
  • #3
Neen87 said:
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:

[tex]|v \times w| = |v||w|\sin\theta[/tex]

where [itex]\theta[/itex] 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 [itex]\theta[/itex] or [itex]\pi - \theta[/itex]. Either way you get the same value for its sine.
 
  • #4
Thanks so much! :-)
 

FAQ: Area of Triangle with Cross Product: Equation Variations

What is the formula for finding the area of a triangle using cross product?

The formula for finding the area of a triangle using cross product is: A = 1/2 * ||a x b||, where a and b are two sides of the triangle and ||a x b|| is the magnitude of their cross product.

How is the cross product used in finding the area of a triangle?

The cross product is used to find the perpendicular vector to two given vectors, which represents the height of the triangle. Multiplying this height by the base (one of the given vectors) and dividing by 2 gives us the area of the triangle.

Are there any other variations of the equation for finding the area of a triangle with cross product?

Yes, there are two other variations of the equation. One is using the dot product instead of the cross product: A = 1/2 * ||a|| * ||b|| * sin(theta), where theta is the angle between a and b. The other variation is using the determinant of a 2x2 matrix formed by the two vectors a and b: A = 1/2 * |a b|, where |a b| is the determinant.

Can the cross product be used to find the area of any triangle?

No, the cross product can only be used to find the area of triangles that have two known sides and the angle between them. It cannot be used for triangles with only three sides given.

Why is the cross product used in finding the area of a triangle?

The cross product is used because it gives us the perpendicular vector to the two given vectors, which is necessary for finding the height of the triangle. It also takes into account the direction of the vectors, ensuring that the area is always positive.

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