Cross product to find the area of a triangle

In summary, the conversation discusses the relationship between the area of a triangle and a parallelogram, with the area of a triangle being half the area of the parallelogram. The conversation also mentions using the cross product of two vectors to find the area of a triangle and asks for clarification on finding the area of a specific triangle.
  • #1
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okay so I know that the area of the triangle is half the area of the parallelogram, ill try using pictures because this is a bit confusing to describe only with words:
for example we have this
http://farside.ph.utexas.edu/teaching/301/lectures/img243.png
and then if we use the cross product of a and b and we find the magnitude of that vector and we divide it by 2 we get the area of the triangle with vertices ADB, so far so good. but i am confused as to how do we find the area of the triangle ADC? my professor said that we can't find it simply by finding the cross product and dividing it by two... anybody can help me with this? :(
thanks!
 
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  • #2
Look at the two triangles ABD and ACD. Geometrically, how would you calculate their areas? Are these areas different?
 

1. What is the cross product method used for finding the area of a triangle?

The cross product method is a mathematical technique used to find the area of a triangle by taking the cross product of two of its sides. This results in a vector perpendicular to the plane of the triangle, and its magnitude is equal to the area of the triangle.

2. How is the cross product calculated to find the area of a triangle?

The cross product of two vectors, a and b, is given by the formula a x b = ||a|| ||b|| sin(θ) n, where ||a|| and ||b|| are the magnitudes of the vectors, θ is the angle between them, and n is a unit vector perpendicular to both a and b.

3. What are the advantages of using the cross product method for finding the area of a triangle?

The cross product method is advantageous because it is a simple and efficient way to calculate the area of a triangle. It also works for any type of triangle, including obtuse and right triangles.

4. Can the cross product method be used for finding the area of any polygon?

No, the cross product method can only be used for finding the area of triangles. For other polygons, different methods, such as the shoelace formula, must be used.

5. How can the cross product method be applied in real-world situations?

The cross product method can be used in various real-world situations, such as calculating the area of a land plot or the surface area of a 3D object. It is also commonly used in engineering and physics to calculate the moment of a force or torque.

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