Final answer:
t(0)=0, r''(0)=24, and r'(t) ⋅ r''(t) = 432e^(3t) - 36e^(-3t) + (12te^(3t) + 4e^(3t))(12e^(3t) + 36te^(3t) + 4e^(3t))
Step-by-step explanation:
To find t(0), we need to substitute t = 0 in the given function r(t). So, substituting t = 0 in r(t), we get:
r(0) = 4e^(3(0))i + 2e^(-3(0))j + 4(0)e^(3(0))k
r(0) = 4i + 2j + 0k
Therefore, t(0) = 0.
To find r''(0), we need to differentiate the given function r(t) twice with respect to t. Differentiating r(t) once, we get:
r'(t) = 12e^(3t)i - 6e^(-3t)j + (12te^(3t) + 4e^(3t))k
Now, differentiating r'(t) with respect to t, we get:
r''(t) = 36e^(3t)i + 6e^(-3t)j + (12e^(3t) + 36te^(3t) + 4e^(3t))k
Substituting t = 0 in r''(t), we get:
r''(0) = 36i + 6j + 16k
Therefore, r''(0) = 36i + 6j + 16k = 24.
To find r'(t) · r''(t), we need to take the dot product of r'(t) and r''(t). So, calculating the dot product, we get:
r'(t) · r''(t) = (12e^(3t)i - 6e^(-3t)j + (12te^(3t) + 4e^(3t))k) · (36e^(3t)i + 6e^(-3t)j + (12e^(3t) + 36te^(3t) + 4e^(3t))k)
r'(t) · r''(t) = 12(36e^(3t)) + 6(-6e^(-3t)) + (12te^(3t) + 4e^(3t))(12e^(3t) + 36te^(3t) + 4e^(3t))
Therefore, r'(t) · r''(t) = 432e^(3t) - 36e^(-3t) + (12te^(3t) + 4e^(3t))(12e^(3t) + 36te^(3t) + 4e^(3t)).