MHB Points of intersection of two vector curves

[carl123]

a) At what point do the curves r1(t) = (t, 2 − t, 35 + t²) and r2(s) = (7 − s, s − 5, s²) intersect?

(x,y,z) =

b) Find their angle of intersection, θ, correct to the nearest degree.

 

[johng1]

a) To find the point(s) of intersection of two curves $r_1(t)$ and $r_2(s)$ you want to find those $t$ and $s$ with $r_1(t)=r_2(s)$; i.e. $t=7-s$, $2-t=s-5$ and $35+t^2=s^2$. For this problem, it turns out there is exactly one $t=t_0$ and $s=s_0$ that satisfy these equations. You can find $t_0$ and $s_0$.

b) The angle of intersection of the two curves is the angle between the two tangent vectors at the intersection point. So find $T_1=r_1^\prime(t_0)$ and $T_2=r_2^\prime(s_0)$. Then with $\theta$ the angle of intersection,
$$\cos(\theta)={T_1\cdot T_2\over ||T_1||\times||T_2||}$$
You can finish from here.