Given,
θ= Independent scalar variable
r(θ)= Position vector
t^= unit tangent vector
n^= unit normal vector
r(θ)=rcosθi^+rsinθj^
n^=∣r(θ)∣r(θ)=rrcosθi^+rsinθj^
=cosθi^+sinθj^
unit tangent vector(t^)=∣r′∣r′(θ)
t^=r−rsinθi^+rcosθj^=−sinθi^+rcosθj^
Now,
n^.t^=(cosθi^+sinθj^)(−sinθi^+rcosθj^)
=−cosθ.sinθ+sinθ.cosθ
=0
Hence, it is prove that n^ is perpendicular to t^ .
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