외적

In Exercises 19–22, the magnitudes of vectors $\vec u$ and $\vec v$ in $\mathbb{R}^3$ are given, along with the angle $\theta$ between them. Use this information to find the magnitude of $\vec u\times\vec v$.

$\left\|\vec u\right\| = 2$, $\left\|\vec v\right\| = 5$, $\theta = 5\pi/6$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle 1,1,2\right\rangle$, $\vec v = \left\langle 2,0,3\right\rangle$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle -2,1,5\right\rangle$, $\vec v = \left\langle -1,3,1\right\rangle$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle 1,2\right\rangle$, $\vec v = \left\langle 2,1\right\rangle$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle 2,0\right\rangle$, $\vec v = \left\langle 0,3\right\rangle$

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(0,0,0)$, $(1,3,-1)$ and $(2,1,1)$.

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(5,2,-1)$, $(3,6,2)$ and $(1,0,4)$.

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(1,1)$, $(1,3)$ and $(2,2)$.

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(3,1)$, $(1,2)$ and $(4,3)$.

In Exercises 31–32, find the area of the quadrilateral with the given vertices. (Hint: break the quadrilateral into 2 triangles.)

Vertices: $(0,0)$, $(1,2)$, $(3,0)$ and $(4,3)$.

In Exercises 31–32, find the area of the quadrilateral with the given vertices. (Hint: break the quadrilateral into 2 triangles.)

Vertices: $(0,0,0)$, $(2,1,1)$, $(-1,2,-8)$ and $(1,-1,5)$.

In Exercises 33–34, find the volume of the parallelepiped defined by the given vectors.

$\vec u = \left\langle 1,1,1\right\rangle$, $\vec v=\left\langle 1,2,3\right\rangle$, $\vec w = \left\langle 1,0,1\right\rangle$

In Exercises 33–34, find the volume of the parallelepiped defined by the given vectors.

$\vec u = \left\langle -1,2,1\right\rangle$, $\vec v=\left\langle 2,2,1\right\rangle$, $\vec w = \left\langle 3,1,3\right\rangle$

In Exercises 35–38, find a unit vector orthogonal to both $\vec u$ and $\vec v$.

$\vec u = \left\langle 1,1,1\right\rangle$, $\vec v=\left\langle 2,0,1\right\rangle$

In Exercises 35–38, find a unit vector orthogonal to both $\vec u$ and $\vec v$.

$\vec u = \left\langle 1,-2,1\right\rangle$, $\vec v=\left\langle 3,2,1\right\rangle$

In Exercises 35–38, find a unit vector orthogonal to both $\vec u$ and $\vec v$.

$\vec u = \left\langle 5,0,2\right\rangle$, $\vec v=\left\langle -3,0,7\right\rangle$

In Exercises 35–38, find a unit vector orthogonal to both $\vec u$ and $\vec v$.

$\vec u = \left\langle 1,-2,1\right\rangle$, $\vec v=\left\langle -2,4,-2\right\rangle$

A bicycle rider applies 150lb of force, straight down, onto a pedal that extends 7in horizontally from the crankshaft. Find the magnitude of the torque applied to the crankshaft.

A bicycle rider applies 150lb of force, straight down, onto a pedal that extends 7in from the crankshaft, making a $30^\circ$ angle with the horizontal. Find the magnitude of the torque applied to the crankshaft.

To turn a stubborn bolt, 80lb of force is applied to a 10in wrench. What is the maximum amount of torque that can be applied to the bolt?