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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