Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a (t) = 2 i + 6t j + 12t2 k, v (0) = i, r (0) = 7 j - 4 k

v (t) = (2t + 1) i + 3t²j + 4t³k

r (t) = (t² + t) i + (t³ + 7) j + (t⁴ - 4) k

Explanation:

a (t) = 2i + 6tj + 12t²k

v (t) = ∫a (t) dt

= ∫ (2i + 6tj + 12t²k) dt

= 2ti + (6t²/2) j + (12t³/3) k + c

= 2ti + 3t²j + 4t³k + c

v (0) = i

i = 0i + 0j + 0k + c

c = i

∴ v (t) = 2ti + 3t²j + 4t³k + i

v (t) = (2t + 1) i + 3t²j + 4t³k

r (t) = ∫ v (t) dt

= i ∫ (2t + 1) dt + 3j ∫ t²dt + 4k ∫ t³dt

= i (2t²/2 + t) + 3j (t³/3) + 4k (t⁴/4) + d

= i (t² + t) + jt³ + t⁴k + d

r (0) = 7j - 4k

0i + 0j + 0k + d = 7j - 4k

d = 7j - 4k

∴ r (t) = (t² + t) i + t³j + t⁴k + 7j - 4k

r (t) = (t² + t) i + (t³ + 7) j + (t⁴ - 4) k