Using vectors, find the angle between the curves y=x^3 and x=y^2 at the point (1,1).
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Parametrize the curves, find the unit tangent vector of each at (1,1), take the dot product, then take the inverse cosine.
r1 = [t, t^3]
r2 = [t^2, t]
r1' = [1, 3t^2]
r2' = [2t, 1]
r1'(1) = [1, 3]
r2'(1) = [2, 1]
r1'(1)/|r1'(1)| = [1/sqrt(10), 3/sqrt(10)]
r2'(1)/|r2'(1)| = [2/sqrt(5), 1/sqrt(5)]
cos(a) = 5 / sqrt(50) = sqrt(2) / 2
a = 45 degrees.
r1 = [t, t^3]
r2 = [t^2, t]
r1' = [1, 3t^2]
r2' = [2t, 1]
r1'(1) = [1, 3]
r2'(1) = [2, 1]
r1'(1)/|r1'(1)| = [1/sqrt(10), 3/sqrt(10)]
r2'(1)/|r2'(1)| = [2/sqrt(5), 1/sqrt(5)]
cos(a) = 5 / sqrt(50) = sqrt(2) / 2
a = 45 degrees.