Question

At what point do the curves r1(t) = t, 3 − t, 35 + t2 and r2(s) = 7 − s, s − 4, s2 intersect?

At what point do the curves r1(t)=<t,3-t,35+t^2> and r2(t) =<7-s,s-4,s^2> intersect?

Also, how do you find the angle of intersection to the nearest degree?

Answer 1

1. r1(t) = r2(s) gives rise to three equations in two variables:

.. t = 7-s

.. 3-t = s-4 ... dependent on the first equation

.. 35+t^2 = s^2

These can be rearranged to

.. s+t = 7

.. s^2 - t^2 = 35 = (s+t)(s-t) = 7(s-t)

From which we determine

.. s = 6, t = 1

and the point of intersection is

.. r1(1) = r2(6) = (1, 2, 36)

2. The cosine of the angle between the curves is the dot product of the normalized direction vectors.

.. r1'(1) = (1, -1, 2*1)

Normalized, this is

.. (1, -1, 2)/√6

and

.. r2'(6) = (-1, 1, 2*6), which normalizes to

.. (-1, 1, 12)/√146

The dot product of these is

.. cos(angle) = ((1)(-1) + (-1)(1) + (2)(12))/((√6)(√146)) = 22/√876 = 11/√219

.. angle = arccos(11/√219) ≈ 41.98° ≈ 42°

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