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#### Dhamnekar Winod

##### Active member

- Nov 17, 2018

- 101

Hello,Continuing this exercise, assume that f'(t) and f''(t) are not parallel. Then $T'(t)\not=0$ so we can define unit principal normal vectorNby

$$N(t)=\frac{T'(t)}{||T'(t)||}$$

Now how to show that $$N(t)=\frac{f'(t)\times (f''(t)\times f'(t))}{||f'(t)||*(||f''(t)\times f'(t)||)}$$

Continuing this execise we can defineunit binormal vectorB$$B(t)=T(t)\times N(t)$$ where $$T(t)=\frac{f'(t)}{||f'(t)||}$$. Note: We have already definedT'(t).

Now how to show that $$B(t)=\frac{f'(t)\times f''(t)}{||f'(t)\times f''(t)||}$$

I want to continue this exercise with one more question related to this question. How does the vectorsT(t), N(t), B(t)form a right-handed system of mutually perpendicular unit vectors (called orthonormal vectors) at each point on the curvef(t)? In the answer to this question, I want to clear explanation about Osculating plane, Normal plane and Rectifying plane.

How can i mark this question (thread) "

**SOLVED"**