First, we need to understand the problem. The curve r(t) is given by three separate components based on the parameter t: sin(πt), cos(3πt), and (t + 2)² + 6. We are asked to find the values of the parameter t such that the curve passes through the point (0, -1, √15).
Moreover, we also need to find the tangent vector(s) at these points, which would involve computing the derivative of each function r1(t) = sin(πt), r2(t) = cos(3πt), r3(t) = (t + 2)² + 6, and evaluating it at the obtained t values.
Step 1: Equating the parts of r(t) to the coordinates of the given point
The given point is (0, -1, √15). If the curve r(t) passes through this point, this means that solving r1(t) = 0, r2(t) = -1, and r3(t) = √15 will give us the desired solution.
Step 2: Solving these equations
Now, we solve these equations to get the values of t.
For the first two cases, unfortunately there are no solutions for t. We can verify this by using the fact that the range of the sin and cos functions is [-1,1] and our equations exceed this range, thus there are no solutions.
Step 3: Compute the derivatives and find the tangent vectors
Since we couldn't find any suitable values of t satisfying all three equations simultaneously, it follows that there are no points on the curve where we could compute the tangent vectors.
Hence, the result is a lack of solutions, both for the values of t and the respective tangent vectors. To be specific, the solution set for this problem is: ([], []).