Given the curve $$r(t) = (1 − 2t)i + (t^2)j + (t^3/2)k, ~~ t > 0~.$$ Find a point on the curve at which the tangent line is parallel to the plane $$5x + y + z − 3 = 0~.$$
I have done everything including making the dot product of the normal of the plane and the direction of the tangent ($r'(t)$) equal to zero but it gives two values of $t$, are both values valid? How do I identify that?
Best Answer
By checking a necessary requirement for $t$, you've reduced the possible values of $t$ to a finite number of cases (two cases to be precise). The next step is now to check for every one of these values of $t$ whether the tangent at $r(t)$ is defined, and if so, indeed parallel to that plane.