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I'm reading this piece from George Cain & James Harod's multivariable calculus material.

Section 4.3, which is about Torsion, says this:

LetR(t) be a vector description of a curve. IfTis the unit tangent andNis the principal unit normal, the unit vectorB=T×Nis called the binormal. Note that the binormal is orthogonal to bothTandN. Let’s see about its derivative dB/ds with respect to arclength s. First, note thatB⋅ B= 1, and soB⋅ (dB/ds) = 0 , which means that being orthogonal toB, the derivative dB/ds is in the plane ofTandN. Next, note thatBis perpendicular to the tangent vectorT, and soB⋅ T= 0 . Thus (dB/ds) ⋅T= 0 . So what have we here? The vector is perpendicular to bothBandT, and so must have the direction ofN(or, of course, -N). This means (dB/ds) = −τN.

The scalar τ is called the torsion.

I don't understand how he deduces dB/ds is perpendicular to T? Where did I get lost?

Following the paragraph, it seems to me that T and N plays quite the same role to B, then suddenly dB/ds is perpendicular to T.

Please enlightent me. Many thanks..