Find the length and sketch the curve. r(t)=[a cos³t,a sin³t], total length.

Calculate the curvature.

$$\mathbf{r}(t)=\cos 3 t \mathbf{i}-4 t \mathbf{j}+\sin 3 t \mathbf{k}$$

Find the first and second derivatives of [4cost, 4sint, 2t].

Find the length and sketch the curve. r(t)=[4 cos t, 4 sin t, 5t] from (4, 0, 0) to (4, 0, 10π)

Sketch the space curve and find its length over the given interval. Vector-Valued Function: r(t) = ⟨2 sin t, 5t, 2 cos t⟩

Let a=[1,1,1], b=[2,3,1], c=[-1,1,0]. Find the angle between:

a - c, b - c

if |p|=6 and |q|=4, what can you say about the magnitude and direction of the resultant? Can you think of an application to robotics?

For vectors $\mathbf{x}, \mathbf{y}, \mathbf{v}, \mathbf{w} \text { in } \mathbf{R}^{3}$ prove the Lagrange identity

$$(\mathbf{x} \times \mathbf{y}) \cdot(\mathbf{v} \times \mathbf{w})=\left|\begin{array}{ll} {\mathbf{x} \cdot \mathbf{v}} & {\mathbf{x} \cdot \mathbf{w}} \\ {\mathbf{y} \cdot \mathbf{v}} & {\mathbf{y} \cdot \mathbf{w}} \end{array}\right|$$

(a) By hand. (b) By computer

This exercise illustrates that poor quality can affect schedules and costs. A manufacturing process has 100 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically, 2% of the components are identified as defective, and the components can be assumed to be independent. (a) If the manufacturer stocks 100 components, what is the probability that the 100 orders can be filled without reordering components? (b) If the manufacturer stocks 102 components, what is the probability that the 100 orders can be filled without reordering components? (c) If the manufacturer stocks 105 components, what is the probability that the 100 orders can be filled without reordering components?

Find the resultant (in components) and its magnitude.

$\mathbf{p}=[3.0,-2], \mathbf{q}=[2,5,1], \mathbf{u}=4 \mathbf{q}$

Let f=xy-yz, v=[2y, 2z, 4x+z], and w=[3z², x²-y², y²]. Find: grad f and f grad f at P:(2, 7, 0)

Determine the magnitude of the resultant force on the indicated area and the location of the center of pressure. Show the resultant force on the area and clearly dimension its location.

Determine the magnitude of the resultant force on the indicated area and the location of the center of pressure. Show the resultant force on the area and clearly dimension its location.