벡터와 공간 전체 문제

A weight of $p$lb is suspended from a chain of length $\ell$ while a constant force of $\vec F_w$ pushes the weight to the right, making an angle of $\theta$ with the vertical, as shown in the figure below.

In Exercises 33–36, a force $\vec F_w$ and length $\ell$ are given. Find the angle $\theta$ and the height the weight is lifted as it moves to the right.

$\vec F_w=1$lb, $\ell = 1$ft, $p = 10$lb

A weight of $p$lb is suspended from a chain of length $\ell$ while a constant force of $\vec F_w$ pushes the weight to the right, making an angle of $\theta$ with the vertical, as shown in the figure below.

In Exercises 33–36, a force $\vec F_w$ and length $\ell$ are given. Find the angle $\theta$ and the height the weight is lifted as it moves to the right.

$\vec F_w=1$lb, $\ell = 10$ft, $p = 1$lb

A weight of $p$lb is suspended from a chain of length $\ell$ while a constant force of $\vec F_w$ pushes the weight to the right, making an angle of $\theta$ with the vertical, as shown in the figure below.

In Exercises 33–36, a force $\vec F_w$ and length $\ell$ are given. Find the angle $\theta$ and the height the weight is lifted as it moves to the right.

$\vec F_w=10$lb, $\ell = 10$ft, $p = 1$lb

The dot product of two vectors is a _____, not a vector.

How are the concepts of the dot product and vector magnitude related?

How can one quickly tell if the angle between two vectors is acute or obtuse?

Give a synonym for “orthogonal.”

In Exercises 5–10, find the dot product of the given vectors.

$\vec u = \left\langle 2,-4\right\rangle$, $\vec v = \left\langle 3,7\right\rangle$

In Exercises 5–10, find the dot product of the given vectors.

$\vec u = \left\langle 5,3\right\rangle$, $\vec v = \left\langle 6,1\right\rangle$

In Exercises 5–10, find the dot product of the given vectors.

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

In Exercises 5–10, find the dot product of the given vectors.

$\vec u = \left\langle 3,5,-1\right\rangle$, $\vec v = \left\langle 4,-1,7\right\rangle$

In Exercises 5–10, find the dot product of the given vectors.

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

In Exercises 5–10, find the dot product of the given vectors.

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

Create your own vectors $\vec u$, $\vec v$ and $\vec w$ in $\mathbb{R}^2$ and show that $\vec u\cdot (\vec v+\vec w) = \vec u\cdot \vec v + \vec u\cdot \vec w$.

Create your own vectors $\vec u$ and $\vec v$ in $\mathbb{R}^3$ and scalar $c$ and show that $c(\vec u\cdot \vec v) = \vec u\cdot (c\vec v)$.

In Exercises 13–16, find the measure of the angle between the two vectors in both radians and degrees.

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

In Exercises 13–16, find the measure of the angle between the two vectors in both radians and degrees.

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

In Exercises 13–16, find the measure of the angle between the two vectors in both radians and degrees.

$\vec u = \left\langle 8,1,-4\right\rangle$, $\vec v = \left\langle 2,2,0\right\rangle$

In Exercises 13–16, find the measure of the angle between the two vectors in both radians and degrees.

$\vec u = \left\langle 1,7,2\right\rangle$, $\vec v = \left\langle 4,-2,5\right\rangle$

In Exercises 17–20, a vector $\vec v$ is given. Give two vectors that are orthogonal to $\vec v$.

$\vec v = \left\langle 4,7\right\rangle$