Checking that a parametric curve is general helix

[demonelite123]

i have the curve a(t) = (3t, 2t², 2t³) and that a'(t) = (3, 4t, 6t²). my textbook tells me to verify that the tangent lines make a constant angle with the line y = 0, z = x so basically the vector (1, 0, 1).

using the definition of the dot product $a * b = |a| |b| cos(\theta)$ i have $cos(\theta) = \frac{3 + 6t^2}{\sqrt{2}\sqrt{9 + 16t^2 + 36t^4}}$

however it doesn't look to me that this is will give a constant angle. the parameter t doesn't seem to cancel out so it wouldn't be constant in this case. have i missed something?

 

[ForMyThunder]

i have the curve a(t) = (3t, 2t², 2t³) and that a'(t) = (3, 4t, 6t²). my textbook tells me to verify that the tangent lines make a constant angle with the line y = 0, z = x so basically the vector (1, 0, 1).

using the definition of the dot product $a * b = |a| |b| cos(\theta)$ i have $cos(\theta) = \frac{3 + 6t^2}{\sqrt{2}\sqrt{9 + 16t^2 + 36t^4}}$

however it doesn't look to me that this is will give a constant angle. the parameter t doesn't seem to cancel out so it wouldn't be constant in this case. have i missed something?

Are you sure it is (3t,2t^2,2t^3) and not (3t,3t^2,2t^3)? Because it would work if it was.

 

[demonelite123]

i noticed that too. I'm guessing its just a typo in the book since changing it to a 3 makes it work out perfectly. thanks for your reply!