How Do You Calculate the Binormal Vector B(t) for a Given Curve?

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In summary, the conversation discusses finding N(t), T(t), and B(t) for the given function r(t) = <t, 4-t^2, 0>. The steps to finding N(t) involve taking the derivative of r(t), dividing it by its length, and then taking the derivative and dividing by its length again. The cross product of T(t) and N(t) should give the binormal vector, but there is an error in the calculation.
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S-C-3-1-3
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Hi! :)

The question is for r (t) =<t, 4-t2, 0>, find N(t),T(t), and B(t).

First I took the derivative of r (t), and divided it by its length to calculate T(t), which is <1/sqrt(1+4t2),-2t/sqrt(1+4t2,0)>. Then I took the derivative of this using the product rule, and divided it by its length to calculate N(t), which is <-2t/sqrt(1+4t2),-1/sqrt(1+4t2),0)>.

Taking the cross product of these two gives the binormalvector, which is <0,0,0>.

Is this done correctly, or does anyone get a different answer?


Thanks in advance! I didn't do all of the work on here, but I hope that's okay, as I have already done the work to find the answer...?
 
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  • #2
Show the steps for finding N(t). You have an error in this.

The cross-product is not zero. Did you take the scalar product?
 
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Related to How Do You Calculate the Binormal Vector B(t) for a Given Curve?

1. What is the binormal vector?

The binormal vector is a mathematical concept used in the study of curves and surfaces in three-dimensional space. It is a vector that is perpendicular to both the tangent vector and the normal vector of a curve or surface at a specific point.

2. How is the binormal vector calculated?

The binormal vector can be calculated by taking the cross product of the tangent vector and the normal vector at a given point on a curve or surface. This results in a vector that is orthogonal, or perpendicular, to both the tangent and normal vectors.

3. What is the significance of the binormal vector?

The binormal vector is important in understanding the orientation and curvature of a curve or surface at a specific point. It is also used in various mathematical and physical applications, such as in the study of motion and forces in physics.

4. How is the binormal vector used in calculus?

In calculus, the binormal vector is used to calculate the curvature of a curve at a given point. It is also used in the calculation of the torsion, or twisting, of a curve in three-dimensional space. These concepts are important in the study of differential geometry and other advanced mathematical topics.

5. Can the binormal vector be visualized?

Yes, the binormal vector can be visualized as a vector that is perpendicular to both the tangent and normal vectors at a specific point on a curve or surface. It can also be represented graphically as a vector that is perpendicular to the curve or surface itself. Visualization of the binormal vector can help in understanding its role in the orientation and curvature of a curve or surface.

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