Answer:
The angle between the curves is 80.27°.
Step-by-step explanation:
Given that
r₁(t) = <2t, t²,t³>
Differentiating with respect to t
r'₁(t) = <2, 2t,3t²>
Since it is intersect at origin.
Then r'₁(0)= <2,0,0> [ putting t=0]
The tangent vector at origin of r₁(t) is
r'₁(0)= <2,0,0>
Again,
r₂(t)= <sin t, sin 5t, 3t>
Differentiating with respect to t
r'₂(t)= <cost, 5 cos 5t, 3>
Since it is intersect at origin.
Then r'₂(0)= <1,5,3> [ putting t=0]
The tangent vector at origin of r₂(t) is
r'₂(0)= <1,5,3>
The angle between the carves is equal to the angle between their tangent.
We know that
[tex]Cos \theta = \frac{r'_1(0).r'_2(0)}{|r'_1(0)||r'_2(0)|}[/tex]
Putting the all values
[tex]Cos \theta = \frac{<2,0,0>.<1,5,3>}{\sqrt{2^2+0^2+0^2}\sqrt{1^2+5^2+3^2}}[/tex]
[tex]\Rightarrow \theta =cos^{-1}(\frac{2}{2\sqrt{35}})[/tex]
⇒θ= 80.27°
The angle between the curves are 80.27°.
