Please help with these questions

Q1:

Determine the equation of the tangent line to the path r(t) = (sin(4t), cos(4t), 1215/12) at t = 1. (Write your solution using the form (*,*,*). Use symbolic notation and fractions where needed. Use the equation of the tangent line such that the point of tangency occurs when t = 1.) tangent line: 1(t) : I Find the velocity and acceleration vectors and the equation of the tangent line for the curve r(t) = ti + e'" j + 7e"k at t = 0. (Use symbolic notation and fractions Where needed. Give your answers in the form (*,*,*).) (Use symbolic notation and fractions Where needed. Give your answers in the form (*,*,*). Use t for the parameter that takes all real values.) Let c1(t) = eS'i + 7 sin(t)j + t4k and egg) = e-5'i + 5 cos(t)j 7t4k. (Enter your solution as a single vector using the vector form (*,*,*). Use symbolic notation and fractions where needed.) d $th (t) + C2(t)] = Let c1(t) = e4'i + 9 sin(t)j + t3k and 02(1) = e-4'i + Zcos(t)j 5t3k. Determine the components of the result, a, b, and c, where %[c1 (t) x 90)] = (a, b, c) (Express numbers in exact form. Use symbolic notation and fractions where needed. NOTE: this problem is computationally challenging.) Find the path c such that c(0) = (0, 12, 15) and c'(t) = (t, e', t2) . (Enter your solution using the vector form (*,*,*). Use symbolic notation and fractions where needed.) c(t) = |