Final answer:
To find the tangent vector at the indicated value of t, we need to take the derivative of the given vector function r(t) and substitute the value of t. The tangent vector at t = 4 is sin(12)i cos(16)j - 12cos(12)cos(16)k - 16sin(12)sin(16)k.
Step-by-step explanation:
To find a tangent vector at the indicated value of t, we first need to find the derivative of the given vector function r(t). The tangent vector is given by the derivative of r(t).
We have r(t) = ti sin(3t)j cos(4t)k. Taking the derivative, we get dr/dt = i * sin(3t)j * cos(4t)k + ti * (3cos(3t))j * cos(4t)k + ti * sin(3t)j * (-4sin(4t))k.
Substituting t = 4, we get dr/dt = i * sin(12)j * cos(16)k + 4i * (3cos(12))j * cos(16)k + 4i * sin(12)j * (-4sin(16))k.
Therefore, the tangent vector at t = 4 is dr/dt = sin(12)i * cos(16)j - 12cos(12)cos(16)k - 16sin(12)sin(16)k.