Vector Introduction

INTRODUCTION

Our world is three dimensional, and we need to develop the tools to describe that three-dimensional world. Those tools are called vectors.

So what exactly are vectors? They are tools to help us better understand all about direction. Technically speaking, vectors are a quantity that has magnitude and direction.

Vector – quantity having both magnitude and direction (but not location)

Scalar – quantity having magnitude only

As an example, let’s say that we start with a distance, 100 m. Not an overly helpful distance at this point. And we know it’s a scalar. How do we make it a vector? Just that easy, we add a direction and it’s now a displacement.

And how about a speed? 50 miles per hour. OK. Now let’s make it a vector. 50 miles per hour east. Because it’s now a vector quantity, we call it velocity.

VECTOR ALGEBRA

Basic vector algebra is actually pretty straightforward, given a few simple rules.

How do we multiply a vector by a scalar quantity? For example, what is 3v ? We just multiply the magnitude of our vector by 3. Or, if our vector is given graphically, i.e. v=<3,2>, 3*v = 3*<3,2> = <9,6>. Pretty easy so far, right?

GRAPHICAL ADDITION

How about vector addition? Well, we can add vectors together graphically or quantitatively. Let’s start out with graphically. To add vectors graphically, we use what’s called the “Tip-to-Tail” Method to determine the resultant vector.

1. Identify each vector by number (1, 2, 3, …)
2. Place the tail of vector 2 on the tip of vector 1, and so on.
3. Draw resultant vector from tail of first to tip of last.

No matter what order you add the vectors, you get the same result. This is because vector addition is commutative.

(NOTE – the direction or magnitude of any vector can’t be changed – only its position in space can be changed)

SUBTRACTION

But what if we want to subtract? Easy. We just use the property that A-B = A+ (-B).

How do we do that? Graphically, we just switch the tip and tail on the vector we’re subtracting (B).

Let’s try a few:



Draw the vectors:

1. A+B
2. A-B
3. 2A-B

Draw the vector: C+D

OK, so hopefully we’re starting to get a picture for what a vector is. Let’s talk in specifics for a bit.

Distance = the total measure of how far an object moves

Displacement = the straight line distance between where we start and where we finish. Requires a direction.

Now, let’s assume a stranger pulls up to you in the parking lot, rolls down their window, and, despite all common sense telling you to duck or run, you assume they’re “not creepy” and help them out. They ask you, “How do I get to the Blue Cross Arena?” How might you answer that?

Distance?

Road directions?

Distance w/direction (AHA! A vector!)

Now, we can discuss the difference between speed and velocity. Serendipitously (my 6-syllable word for the day), the first letters of the items match:

Speed = the rate at which an object’s distance changes  (scalar)

Velocity = the rate at which an object’s displacement changes. HAS DIRECTION (vector)

Where it gets tricky, though, is when you realize that both quantities use the same formula: v=d/t

With these questions, you’re going to have to keep an eye on the context. Am I asking for a speed, or a velocity? Do I ask for your distance traveled, or your displacement?

(One hint, vectors are often depicted in bold, or with an arrow over them)