Tangent on given curve makes constant angle with Z axis.

Question

Show that tangent at any point on the curve x = (e^t)cost, y = (e^t)sint, z=e^t makes constant angle with Z axis.

Answer 1

Vector r=(e^tcost, e^tsint, e^t), the i, j, k unit vectors for x, y, z values are understood by the brackets.

Derivative r'=(e^t(cost-sint), e^t(cost+sint), e^t).

The unit tangent vector is given by r'/r where r is the scalar value of the right-hand side=sqrt(x^2+y^2+z^2), which is sqrt(e^2t(cos^2t+sin^2t-2sintcost)+e^2t(cos^2t+sin^2t+2sintcost)+e^2t)=sqrt(e^2t(1+1+1)=e^tsqrt(3).

The unit tangent vector is ((cost-sint, cost+sint, 1)e^t)/(e^tsqrt(3))=(cost-sint, cost+sint, 1)/sqrt(3).

The cosine of the angle of the tangent is the dot (scalar) product of the unit tangent vector with k=1/sqrt(3). Since this does not involve t it is constant for all values of t.