The Dot Product

Determine each of the following:

1. The length of the vector $\vu=\langle 1, 2, -3 \rangle$ using the dot product.

2. The angle between the vectors $\vu =\langle 1, 2 \rangle$ and $\vv = \langle 4, -1 \rangle$ to the nearest tenth of a degree.

3. The angle between the vectors $\vy =\langle 1, 2, -3 \rangle$ and $\vz = \langle -2, 1, 1 \rangle$ to the nearest tenth of a degree.

4. If the angle between the vectors $\vu$ and $\vv$ is a right angle, what does the expression $\vu \cdot \vv = \vecmag{\vu} \vecmag{\vv} \cos(\theta)$ say about their dot product?

5. If the angle between the vectors $\vu$ and $\vv$ is acute—that is, less…

1. Determine the work done by a 25 pound force acting at a $30^{\circ}$ angle to the direction of the object's motion, if the object is pulled 10 feet.

2. Determine if more work or less work is done if the angle to the direction of the object's motion is $60^\circ$.

Let $\vv = \langle -2, 5 \rangle$ in $\R^2$, and let $\vy = \langle 0, 3, -2 \rangle$ in $\R^3$.

1. Is $\langle 2, -1 \rangle$ perpendicular to $\vv$? Why or why not?

2. Find a unit vector $\vu$ in $\R^2$ such that $\vu$ is perpendicular to $\vv$. How many such vectors are there? Justify your answers.

3. Is $\langle 2, -1, -2 \rangle$ perpendicular to $\vy$? Why or why not?

4. Find a unit vector $\vw$ in $\R^3$ such that $\vw$ is perpendicular to $\vy$. How many such vectors are there?Justify your answers.

5. Let $\vz = \langle 2, 1, 0 \rangle$. Find a unit vector $\vr$ in $\R^3$ such that $\vr$ is perpendicular to…

Let $\vu$ and $\vv$ be vectors in $\R^5$ with $\vu \cdot \vv = -1$, $\Vert \vu \Vert = 2$, $\Vert \vv \Vert = 3$. Use the properties of the dot product to find each of the following.

1. $\vu \cdot 2\vv$

2. $\vv \cdot \vv$

3. $(\vu + \vv) \cdot \vv$

4. $(2\vu+4\vv) \cdot (\vu - 7\vv)$

5. $\Vert\vu\Vert \Vert\vv\Vert \cos(\theta)$, where $\theta$ is the angle between $\vu$ and $\vv$

6. $\theta$

One of the properties of the dot product is that $(\vu+\vv) \cdot \vw = (\vu \cdot \vw) + (\vv \cdot \vw)$. That is, the dot product distributes over vector addition on the right. Here we investigate whether the dot product distributes over vector addition on the left.

1. Let $\vu = \langle 1,2,-1 \rangle$, $\vv = \langle 4,-3,6 \rangle$, and $\vw = \langle 4,7,2 \rangle$. Calculate What do you notice?

2. Use the properties of the dot product to show that in general for any vectors $\vx$, $\vy$, and $\vz$ in $\R^n$.

When running a sprint, the racers may be aided or slowed by the wind. The wind assistance is a measure of the wind speed that is helping push the runners down the track. It is much easier to run a very fast race if the wind is blowing hard in the direction of the race. So that world records aren't dependent on the weather conditions, times are only recorded as record times if the wind aiding the r…