91Ó°ÊÓ

Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the following parameterized curves. $$\mathbf{r}(t)=\langle 2 t, 4 \sin t, 4 \cos t\rangle$$

Find equations of the following lines. The line through (0,0,0) and (1,2,3)

How do you compute \(|\overrightarrow{P Q}|\) from the coordinates of the points \(P\) and \(Q ?\)

Compute \(\mathbf{u} \cdot \mathbf{v}\) if \(\mathbf{u}\) and \(\mathbf{v}\) are unit vectors and the angle between them is \(\pi / 3\).

Arc length calculations Find the length of the following two and three- dimensional curves. $$\mathbf{r}(t)=\langle\cos t+t \sin t, \sin t-t \cos t\rangle, \text { for } 0 \leq t \leq \pi / 2$$

Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $$\mathbf{r}(t)=\left\langle t^{2}+3, t^{2}+10, \frac{1}{2} t^{2}\right\rangle, \text { for } t \geq 0$$

Compute \(|\mathbf{u} \times \mathbf{v}|\) if \(\mathbf{u}\) and \(\mathbf{v}\) are unit vectors and the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(\pi / 4\)

Compute \(|\mathbf{u} \times \mathbf{v}|\) if \(|\mathbf{u}|=3\) and \(|\mathbf{v}|=4\) and the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(2 \pi / 3\)

Arc length calculations Find the length of the following two and three- dimensional curves. $$\mathbf{r}(t)=\langle\cos t+\sin t, \cos t-\sin t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Compute \(\mathbf{u} \cdot \mathbf{v}\) if \(\mathbf{u}\) is a unit vector, \(|\mathbf{v}|=2,\) and the angle between them is \(3 \pi / 4\).