How Do Perpendicular Vectors Around a Triangle Sum to Zero?

In summary, the conversation discusses showing that the sum of three vectors perpendicular to the sides of a triangle and pointing towards the exterior of the triangle is equal to 0. The dot product formula and the concept of circulation integrals are mentioned as potential tools for proving this. The solution involves applying a 90 degree rotation and using the dot product formula to show that the sum of the three vectors is indeed equal to 0.
  • #1
skate_nerd
176
0

Homework Statement



So a, b, and c are points in the plane. Let nab, nbc, and nca be vectors perpendicular to ab(vector), bc(vector), and ca(vector) respectively, and point towards the exterior of the triangle abc. Also, |nab|=|ab(vector)|, |nbc|=|bc(vector)|, and |nca|=|ca(vector)|. Show that nab+nbc+nca=0.

Homework Equations



I'm guessing that the formula for the dot product will be used, and that nab(dot)ab=0, and same for the other two vector combinations.

The Attempt at a Solution



Also, we have been learning about circulation integrals and line integrals. Not really sure if that proves much, but I know that the circulation around this triangle would be equal to 0 also, so there's something. Not sure really what else I have to go on though. I'm not very well versed in proofs, and my calc 3 teacher sure loves making us do them.
 
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  • #2
You know that

ab + bc + ca = 0

Now simply apply a rotation by 90 degrees
 

Related to How Do Perpendicular Vectors Around a Triangle Sum to Zero?

1. What is the definition of vector circulation?

Vector circulation is a measure of the flow of a vector field around a closed path or curve. It is mathematically represented by the line integral of the vector field along the closed path.

2. How is vector circulation different from scalar circulation?

Vector circulation takes into account both the magnitude and direction of the vector field, while scalar circulation only considers the magnitude. Vector circulation is also represented by a line integral, whereas scalar circulation is represented by a surface integral.

3. What is the significance of vector circulation in fluid dynamics?

In fluid dynamics, vector circulation plays a crucial role in understanding the movement and behavior of fluids. It helps in determining the strength and direction of vortices, which are important in the formation of turbulence and other flow patterns.

4. How is vector circulation calculated?

Vector circulation is calculated using the line integral of the vector field along a closed path or curve. This involves breaking down the path into small segments and calculating the dot product of the vector field and the tangent vector at each segment. The sum of these dot products gives the total vector circulation.

5. What are some real-world applications of vector circulation?

Vector circulation has many practical applications, including weather forecasting, aerodynamics, and fluid engineering. It is also used in the design of aircraft wings, turbines, and other structures that interact with fluid flow.

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