Vector Calculus Principle Normal Vector and Binomial Vector

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If T'(t) does not equal 0, it follows that N(t) = T'(t)/||T'(t)||is normal to t(t); is called the principle normal vector. Let a third unit vector that is perpendicular to both T and N be defined by B = T x N; B is called the binomial vector. Together, T, N and B form a right-handed system of mutually orthogonal vectors that may be thought of as moving along a path as shown below.

Show that

a) dB/dt . B = 0
b) dB/dt . T = 0
c) dB/dt = constat * N

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(A) Show dB/dt . B = 0

B . B = 1

Differentiate w.r.t: dB/dt . B + B . dB/dt = 0

2 dB/dt . B = 0

dB/dt . B = 0

(B) Show dB/dt . T = 0

B is normal to T

B . T = 0

dB/dt . T + B . dT/dt = ...

Solution Summary

I have used vector calculus to prove the given identities. Solution is in a 2-page word document. I have provided each and every step in this proof, so that the students who decide to down load this answer will understand the process well. They also will be able to use similar techniques in other problems encountered in vector calculus.

Let p1 = (4,0,4), p2 = (2,-1,8), and p3 = (1,2,3).
a) Show that the three points define a right triangle. Hint the difference between two vertices is a vector whose direction coincides with that of a triangle side, and a pair of such vectors must be orthogonal in order for the triangle to be a right triangle.
b) Specify

The medial triangle of a triangle ABC is the triangle whose vertices are located at the midpoints of the sides AB, AC, and BC of triangle ABC. From an arbitrary point O that is not a vertex of triangle ABC, you may take it as a given fact that the location of the centroid of triangle ABC is the vector (vector OA + vector OB + ve

Please answer eight (8) calculus problems. Please show as much works as possible for every problem. The problems are posted in the following website:
http://www.netprofitspro.com/math.html

The eastward component of vector A is equal to the westward component of vector B and their northward components are equal. Which one of the following statements is correct for these two vectors?
Choices:
Vector A is parallel to vector B
Vector A is anti-parallel to vector B
The magnitude of vector A is equal to the mag

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Let W be the set of all 2x2 matrices of the form [a -b]. Then under the usual matrix operations W is
[b -a]
1. Not a vector space
2. a vector space of dimension 1.
3. a vector space of dimension 2.
4. a vector space of dimension 3.
5. a vecto

Consider the vector field F = (x^2 + y^2, 8xy). Compute the line integrals and , where c1(t) = (t, t^2) and c2(t) =(t, t) for 0<=t<=1. Can you decide from your answers whether or not F is a gradient vector field? Why or why not?

Vector A is 3.00 units in length and points along the positive x-axis. Vector B is 4.00 units in length and points along the negative y-axis. Use the graphical methods to find the magnitude and direction of the following vector:
B.) B-A