Math 214 - Multivariable Calculus - Prep

Task 3.1

Suppose for a short time that a rover follows a path given by $(x, y) = (1t + 3, −2t + 4)$. This is the same as writing $(x, y) = (1, −2)t + (3, 4)$.

Construct a plot that shows the location of the rover at time $t = 0, 1, 2$, and add some arrows as well as a line to illustrate the rover’s path.
What is the speed of the rover? (you may assume that distances are in meters, and time is in minutes).
What is the rover's velocity (hint, this should be a vector)?
When we write the path in the form $(x, y) = (1, −2)t + (3, 4)$, what do the quantities (1, −2) and (3, 4) have to do with the path?
The rover is no longer on flat ground, rather is sitting at point $P = (0, 2, 3)$. It starts to climb in the direction $\vec v = \langle 1, −1, 2\rangle$.
- Write a vector equation $(x, y, z) = (?, ?, ?)$ for the line that passes through the point $P$ and is parallel to $\vec v$.
- Generalize your work to give an equation of the line that passes through the point $P = (x_1 , y_1 , z_1)$ and is parallel to the vector $\vec v = (v_1 , v_2 , v_3 )$.

Task 3.2

Suppose the Curiosity rover travels in a circular path given by the parametric curve $\vec r(t) = (3 \cos t, 3 \sin t)$.

Graph the curve $\vec r$ (you should obtain a circle) and make sure you designate the direction in which the rover is traveling.
Compute both $\ds \frac{d \vec r}{dt}$ and $\ds\frac{d^2\vec r}{dt^2}$.
Locate the point on your graph that the rover is at when $t = \pi/2$. How would you describe the velocity and acceleration of the rover at this point? Compute both $\frac{d\vec r}{dt}(\frac{\pi}{2})$ and $\frac{d^2\vec r}{dt^2}(\frac{\pi}{2})$, and confirm that these vectors do indeed provide the acceleration and velocity of the rover at $t = \pi/2$.
Let's swap to the time $t = \pi/4$. On your graph, draw the vectors $\frac{d\vec r}{dt}(\frac{\pi}{4})$ and $\frac{d^2\vec r}{dt^2}(\frac{\pi}{4})$ with their tail placed on the curve at $\vec r(\frac{\pi}{4})$. These vectors are the velocity and acceleration.
Give a vector equation of the tangent line to this curve at $t = \pi/4$.

Task 3.3

Suppose a heavy box needs to be lowered down a ramp. The box exerts a downward force of say 200 Newtons (gravity), which we could write in vector notation as $\vec F=\left<0,-200\right>$. If the ramp was placed so that the box needed to be moved right 6 m, and down 3 m, then we'd need to get from the origin $(0,0)$ to the point $(6,-3)$. This displacement can be written as $\vec d=\left<6,-3\right>$. The force $\vec F$ acts straight down, rather than parallel to the displacement. Let's find out how much of the force $\vec F$ acts in the direction of the displacement. We are going to break the force $\vec F$ into two components, one component in the direction of $\vec d$, and another component orthogonal to $\vec d$. The component of the force that is parallel to $\vec d$ is useful in understanding energy computations. The component of the force that is orthogonal to $\vec d$ is useful in understanding surface friction.

We want to write $\vec F$ as the sum of two vectors $\vec F = \vec w+\vec n$, where $\vec w$ is parallel to $\vec d$ and $\vec n$ is orthogonal to $\vec d$. Since $\vec w$ is parallel to $\vec d$, we can write $\vec w=c\vec d$ for some unknown scalar $c$. This means that $\vec F=c\vec d+\vec n$.

Start by drawing a picture that shows how $\vec F$, $\vec d$, $\vec w$, and $\vec n$ are related.
Use the fact that $\vec n$ is orthogonal to $\vec d$ to show that $\ds c = \frac{\vec F\cdot \vec d}{\vec d\cdot \vec d}$. [Hint: Dot each side of $\vec F=c\vec d+\vec n$ with $\vec d$ and distribute. You'll need to use the fact that $\vec n$ and $\vec d$ are orthogonal to remove $\vec n\cdot \vec d$ from the problem.]
Now that we have a formula for $c$, use that formula to show that $\vec w = c\vec d = (80,-40)$. We call this the projection of $\vec F$ onto $\vec d$ (or the component of $\vec F$ that is parallel to $\vec d$), and write $$\text{proj}_{\vec d}\vec F = \vec F_{\parallel \vec d}= \left(\frac{\vec F\cdot \vec d}{\vec d\cdot \vec d}\right)\vec d.$$
Obtain a formula for $\vec n$, the component of the force that is orthogonal to $\vec d$. This is sometimes written as $\vec F_{\perp \vec d}$.

Task 3.4

The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below. Remember that you don't have to do all of the problems listed below, rather do a few from sections that you feel you need more practice with.

Equations of lines: Section 2.5, checkpoint 2.43 and exercises 243 - 250.
Derivatives of Vector Valued functions: Section 3.2, checkpoint 3.5 and exercises 41-54.
Projection practice: section 2.3, checkpoint 2.27 and exercises 167-172

Big View
Prep Day 3 Task 3.1 Suppose for a short time that a rover follows a path given by $(x, y) = (1t + 3, −2t + 4)$. This is the same as writing $(x, y) = (1, −2)t + (3, 4)$. Construct a plot that shows the location of the rover at time $t = 0, 1, 2$, and add some arrows as well as a line to illustrate the rover’s path. What is the speed of the rover? (you may assume that distances are in meters, and time is in minutes). What is the rover's velocity (hint, this should be a vector)? When we write the path in the form $(x, y) = (1, −2)t + (3, 4)$, what do the quantities (1, −2) and (3, 4) have to do with the path? The rover is no longer on flat ground, rather is sitting at point $P = (0, 2, 3)$. It starts to climb in the direction $\vec v = \langle 1, −1, 2\rangle$. Write a vector equation $(x, y, z) = (?, ?, ?)$ for the line that passes through the point $P$ and is parallel to $\vec v$. Generalize your work to give an equation of the line that passes through the point $P = (x_1 , y_1 , z_1)$ and is parallel to the vector $\vec v = (v_1 , v_2 , v_3 )$. Task 3.2 Suppose the Curiosity rover travels in a circular path given by the parametric curve $\vec r(t) = (3 \cos t, 3 \sin t)$. Graph the curve $\vec r$ (you should obtain a circle) and make sure you designate the direction in which the rover is traveling. Compute both $\ds \frac{d \vec r}{dt}$ and $\ds\frac{d^2\vec r}{dt^2}$. Locate the point on your graph that the rover is at when $t = \pi/2$. How would you describe the velocity and acceleration of the rover at this point? Compute both $\frac{d\vec r}{dt}(\frac{\pi}{2})$ and $\frac{d^2\vec r}{dt^2}(\frac{\pi}{2})$, and confirm that these vectors do indeed provide the acceleration and velocity of the rover at $t = \pi/2$. Let's swap to the time $t = \pi/4$. On your graph, draw the vectors $\frac{d\vec r}{dt}(\frac{\pi}{4})$ and $\frac{d^2\vec r}{dt^2}(\frac{\pi}{4})$ with their tail placed on the curve at $\vec r(\frac{\pi}{4})$. These vectors are the velocity and acceleration. Give a vector equation of the tangent line to this curve at $t = \pi/4$. Task 3.3 Suppose a heavy box needs to be lowered down a ramp. The box exerts a downward force of say 200 Newtons (gravity), which we could write in vector notation as $\vec F=\left<0,-200\right>$. If the ramp was placed so that the box needed to be moved right 6 m, and down 3 m, then we'd need to get from the origin $(0,0)$ to the point $(6,-3)$. This displacement can be written as $\vec d=\left<6,-3\right>$. The force $\vec F$ acts straight down, rather than parallel to the displacement. Let's find out how much of the force $\vec F$ acts in the direction of the displacement. We are going to break the force $\vec F$ into two components, one component in the direction of $\vec d$, and another component orthogonal to $\vec d$. The component of the force that is parallel to $\vec d$ is useful in understanding energy computations. The component of the force that is orthogonal to $\vec d$ is useful in understanding surface friction. We want to write $\vec F$ as the sum of two vectors $\vec F = \vec w+\vec n$, where $\vec w$ is parallel to $\vec d$ and $\vec n$ is orthogonal to $\vec d$. Since $\vec w$ is parallel to $\vec d$, we can write $\vec w=c\vec d$ for some unknown scalar $c$. This means that $\vec F=c\vec d+\vec n$. Start by drawing a picture that shows how $\vec F$, $\vec d$, $\vec w$, and $\vec n$ are related. Use the fact that $\vec n$ is orthogonal to $\vec d$ to show that $\ds c = \frac{\vec F\cdot \vec d}{\vec d\cdot \vec d}$. [Hint: Dot each side of $\vec F=c\vec d+\vec n$ with $\vec d$ and distribute. You'll need to use the fact that $\vec n$ and $\vec d$ are orthogonal to remove $\vec n\cdot \vec d$ from the problem.] Now that we have a formula for $c$, use that formula to show that $\vec w = c\vec d = (80,-40)$. We call this the projection of $\vec F$ onto $\vec d$ (or the component of $\vec F$ that is parallel to $\vec d$), and write $$\text{proj}_{\vec d}\vec F = \vec F_{\parallel \vec d}= \left(\frac{\vec F\cdot \vec d}{\vec d\cdot \vec d}\right)\vec d.$$ Obtain a formula for $\vec n$, the component of the force that is orthogonal to $\vec d$. This is sometimes written as $\vec F_{\perp \vec d}$. Task 3.4 The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below. Remember that you don't have to do all of the problems listed below, rather do a few from sections that you feel you need more practice with. Equations of lines: Section 2.5, checkpoint 2.43 and exercises 243 - 250. Derivatives of Vector Valued functions: Section 3.2, checkpoint 3.5 and exercises 41-54. Projection practice: section 2.3, checkpoint 2.27 and exercises 167-172 Edit Page History Source Attach File Backlinks List Group Page last modified on September 15, 2022, at 01:55 PM
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Day 3

Task 3.1

Task 3.2

Task 3.3

Task 3.4