(a) Show that d B / d s is perpendicular to B . (b) Show that d B / d s is perpendicular to T . (c) Deduce from parts (a) and (b) that d B / d s = − τ ( s ) N for some number τ ( s ) called the torsion of the curve. (The torsion measures the degree of twisting of a curve.) (d) Show that for a plane curve the torsion is τ ( s ) = 0 .
(a) Show that d B / d s is perpendicular to B . (b) Show that d B / d s is perpendicular to T . (c) Deduce from parts (a) and (b) that d B / d s = − τ ( s ) N for some number τ ( s ) called the torsion of the curve. (The torsion measures the degree of twisting of a curve.) (d) Show that for a plane curve the torsion is τ ( s ) = 0 .
Solution Summary: The author explains that two vectors are perpendicular to each other if their dot product is 0.
(c) Deduce from parts (a) and (b) that
d
B
/
d
s
=
−
τ
(
s
)
N
for some number
τ
(
s
)
called the torsion of the curve. (The torsion measures the degree of twisting of a curve.)
(d) Show that for a plane curve the torsion is
τ
(
s
)
=
0
.
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