The acceleration of an object after t seconds is given by the vector-valued functiona→(t)=⟨cos⁡(8t), sin⁡(8t), −5⟩If the object's initial position is (−9, −3, 10), and the object's initial velocity is ⟨1, 0, 1⟩, find a vector-valued function r→(t) representing the object's position at time t.

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The acceleration of an object after t seconds is given by the vector-valued functiona→(t)=⟨cos⁡(8t), sin⁡(8t), −5⟩If the object&#039;s initial position is (−9, −3, 10), and the object&#039;s initial velocity is ⟨1, 0, 1⟩, find a vector-valued function r→(t) representing the object&#039;s position at time t.

George Morin · Accepted Answer

Step 1  a(t)=(cos⁡(8t), sin⁡(8t). −5)  to find the velocity equation, take integration  v(t)=∫a(t)dt  v(t)=∫(cos⁡(8t), sin⁡(8t), −5)dt  1) v(t)=(18sin⁡(8t)+x, −18cos⁡(8t)+y, −5t+z)  Step 2  here initial velocity is v(0)=(1, 0, 1)  (1, 0, 1)=(18sin⁡(8×0)+x, −18cos⁡(8×0)+ym −5×0+z)  (1, 0, 1)=(0+x, −18+y, z)  (1, 0, 1)=(x, −18+y, z)  (x, −18+y, z)=(1, 0, 1)  (x, y, z)=(1, 18, 1) put it back in equation 1  Step 3  v(t)=(18sin⁡(8t)+x, −18cos⁡(8t)+y, −5t+z)  v(t)=(18sin⁡(8t)+1, −18cos⁡(8t)+18, −5t+1)  to find the position equation, take integration  r(t)=∫v(t)dt

The acceleration of an object after t seconds is given

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