1.3 Vectors

Vectors

Introduction

Representations of Vectors

Equality of Vectors

Operations on Vectors

Postscript is a programming language whose primary purpose is to specify how to generate text or graphics. The following is a simple set of Postscript commands that produces the triangle in the plane with vertices $(0,0)$, $(1,1)$, and $(1,-1)$: (0,0) moveto (1,1) lineto stroke (1,-1) lineto stroke (0,0) lineto stroke The process described by these commands is - tell Postscript to start at the point $(0,0)$, - draw a line from the point $(0,0)$ to the point $(1,1)$ (this is what the line to and stroke commands do), - draw lines from $(1,1)$ to $(1,-1)$, and - draw from $(1,-1)$ back to the origin. Each of these commands encodes two important pieces of information: a direction in which to move and a distance to move. Mathematically, we can capture this information succinctly in a vector. To do so, we record the movement on the map in a pair $\langle x, y \rangle$, where $x$ is the horizontal displacement and $y$ the vertical displacement from one point to another. This pair $\langle x, y \rangle$ is a vector, and we call each number in the pair a component. As an example, the vector from the origin to the point $(1,1)$ is represented by $\langle 1,1 \rangle$ while the vector from the point $(1,1)$ to the origin is represented by $\langle -1,-1 \rangle$.

What is the vector $\vv_1 = \langle x , y \rangle$ that describes the displacement from the point $(1,1)$ to the point $(1,-1)$?

$\overrightarrow{OA}= \langle\, ,\, ,\, \rangle$

$\overrightarrow{OB}= \langle\, ,\, ,\, \rangle$

$\overrightarrow{OC}= \langle\, ,\, ,\, \rangle$

$\overrightarrow{AB}= \langle\, ,\, ,\, \rangle$

$\overrightarrow{AC}= \langle\, ,\, ,\, \rangle$

$\overrightarrow{BC}= \langle\, ,\, ,\, \rangle$

Your friend told you they would be staying 3 km east and 4 km north of the main parking lot. You drive to the parking lot and park next to your friend's car, then hike 3 km east and 4 km north. You get to the location you expected your friend to be at, but you don't find your friend and call the ranger station. The ranger station says they think your friend is 2 km west and 1 km north of your current location. We will denote the location of the car with point $C$, your friend's originally anticipated location as point $A$, and the ranger's suggested location as point $S$.

On the grid below, we have labeled point $C$. Draw the location of each of the other points $A$ and $S$.

Using your picture from the previous task, give the components of the following vectors: -$\overrightarrow{CA}$-$\overrightarrow{CS}$-$\overrightarrow{AS}$

In the context of this problem, explain why you can add the horizontal components of $\overrightarrow{CA}$ and $\overrightarrow{AS}$ to get the horizontal component of $\overrightarrow{CS}$. Write a sentence or two about why this argument should work for the vertical components as well.

After hiking to the location suggested by the ranger, you still don't see your friend and call the ranger station again. The regional manager of rangers answers this time. She says the first ranger made a mistake in their navigation. They sent a drone to the location of your car and the drone spotted your friend along the same direction from your car to point $A$, but your friend is three times as far away from your car as you were when you were at point $A$.

In order to avoid confusion or any other mistakes, you want to compute the vector from your car to the drone's suggested location, which we will call $D$. What are the components of $\overrightarrow{CD}$? Write a sentence or two to compare this vector to $\overrightarrow{CA}$.

So we need to get from point $S$ to point $D$. Explain how to use the components of $\overrightarrow{CS}$ and $\overrightarrow{CD}$ to find the components of $\overrightarrow{SD}$.

You find your friend at point $D$ and want to take them back to your car to go home. Explain how you can use the components of $\overrightarrow{CD}$ to give directions from point $D$ back to your car.