Question

Find d/dt [r(t) · u(t)] and d/dt [r(t) ⨯ u(t)] in two different ways. r(t) = cos(t)i sin(t)j tk, u(t) = j tk.

(a) d/dt [r(t) · u(t)] (i) find the product first, then differentiate. (ii) apply the properties of the derivative.
(b) d/dt [r(t) ⨯ u(t)] (i) find the product first, then differentiate. (ii) apply the properties of the derivative.

Answer 1

To find the derivative of r(t) · u(t), first find the product of r(t) and u(t) and then differentiate. The derivative is cos(t). To find the derivative of r(t) ⨯ u(t), find the product of r(t) and u(t) and then differentiate. The derivative is (cos(t)) j - (t)(sin(t)) j + (cos(t)) k.

Step-by-step explanation:

To find the derivative of r(t) · u(t), first we need to find the product of r(t) and u(t). We can do this by multiplying the corresponding components of r(t) and u(t) together. The dot product is given by r(t) · u(t) = (cos(t))(0) + (sin(t))(1) + (t)(0) = sin(t). Next, we can differentiate the dot product with respect to time. The derivative of sin(t) with respect to t is cos(t), so the derivative of r(t) · u(t) is cos(t).

To find the derivative of r(t) ⨯ u(t), we need to find the product of r(t) and u(t) first. We can do this by using the cross product formula. The cross product is given by r(t) ⨯ u(t) = (0)(tk) - (cos(t))(0) i + (cos(t))(tk) j = t(cos(t)) j - (sin(t)) k. Next, we can differentiate the cross product with respect to time. The derivative of t(cos(t)) j - (sin(t)) k with respect to t is (cos(t)) j - (t)(sin(t)) j + (cos(t)) k, so the derivative of r(t) ⨯ u(t) is (cos(t)) j - (t)(sin(t)) j + (cos(t)) k.