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This isn't a homework question, it's for deeper understanding. So I'm learning about unit normal/tangent vectors and the curvature of a curve. I have a few questions/points.

1) So my book states that we can express acceleration as a linear combination of the acceleration in the same direction as T (the tangent vector of the curve), and the acceleration in the normal/perpendicular direction (direction of N, the normal vector).

--- One thing I'm wondering is: Say we are dealing with a three-dimensional curve, r(t)..

--- The tangent vector to the curve is perpendicular to the position vector; wouldn't the unit normal

--- vector just be parallel to the position vector? Since the unit normal vector is perpendicular to the tangent vector.

--- In that case, could we not just take the derivatives of the position vector and be able to construct the unit tangent and unit normal vectors from that by dividing by their magnitude? Or is that what my book is saying, just in a very convoluted way?

2) I'm taking a Linear Algebra course right now, and I feel as though I can translate what I'm learning in Calculus to what I've learned in Lin. Alg., but I'm not quite there right now.

--- An alternative formula for figuring out the curvature is given in terms of the magnitude of the cross product of v, the velocity vector, and acceleration, the derivative of the velocity vector, divided by the magnitude of the velocity cubed.

--- That sounds like we want a vector perpendicular to v and a divided by some scalar. Is there another way to find the curvature, then?

--- Like taking a vector, removing its projections onto v and a? That would get a vector orthogonal to both, I could then take its magnitude and divide it by something?

I'm a bit lost in trying to integrate my knowledge of Calculus with my knowledge of Linear Algebra. If I take the second derivative of the position function, r(t), aren't the components of it orthogonal? If not, can I just make an orthogonal basis, and then work from there? My head is spinning a bit. :)

Thanks,

ModestyKing

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# Relation between unit tangent/normal vectors, curvature, and Lin. Alg.

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