Understanding Vectors: A Guide to Motion in Physics

I'm here in Blue Ridge, Georgia. We're gonna do some sightseeing, some geocaching,

and learn how to resolve vectors using graphs. [train whistle blows] ♪♪

Vectors are the way we measure motion in physics, showing magnitude and direction. For instance, they can represent how a quarterback

throws a pass downfield. Magnitude is how hard and far he threw the ball, and the direction hopefully is downfield.

Or vectors can tell a pilot how fast and in what direction to fly the plane to overcome a headwind.

Without vectors, we'd have a really hard time describing motion, and that's key the physics we're learning about

in this course. So let's use how this train moves along a track to graph our first vector on a number line.

Trains only go back and forth on a track, right? You can use a number line to draw movement in one dimension, either up or down,

or right or left. So if this train goes 50 meters along the track, here's how we graph that.

This is what a number line looks like. A straight line with numbers that represent actual points. When you're dealing with this kind of graph,

the plus and minus signs indicate direction. You choose which sign means which direction. Remember, they can mean either up or down,

or left or right. So plus could be to the right, and minus to the left. Or, plus could be up and minus down,

or vice versa. So, the train went 50 meters from the platform, which is our origin, in the positive direction,

which is to the right. To graph that, I would mark off the points first, like this.

Now suppose the train went another 30 meters further down the track. With a number line, it is easy to see

that we add the new distance. But, since we're using vectors, we're going to discuss resultants.

That's the result of adding a second vector to the first. To get a resultant, you can add the vectors using the tip-to-tail method.

The tip-to-tail method works like this. Draw a vector using an arrow to represent it. The tip is here, and the tail is there.

Simple, right? So let's graph something on a number line and add our vectors using tip-to-tail.

If we start at the origin, the 0 meter mark here, and we draw our vector to the positive 3 mark, how far did we go?

Three meters, right? Right. Then, if we go another 5 meters in the positive direction, to graph that we draw an arrow,

starting at the positive 3 meter mark, five places in the positive direction, like this. They are tip-to-tail, see?

Now add the vectors to get the resultant. And we have positive 8 meters. And that's the magnitude of the resultant vector.

[train whistle blows] So that's a simple line graph, representing movement in one direction in one dimension.

That's not too bad. Ready for more? Now, we're ready to resolve some 2-D vectors graphically. And to do that,

we're gonna head out to do some geocaching. I have my clues right here. Great. I go north, then northeast, then west.

And, found it! Geocaching is following location clues, kind of like a treasure hunt,

to find objects that other people have left for you. Like this. All right, I'll mark it

and I'll put it back so the next person can find it. Now my producers left another cache near here. Let's go look for it.

Okay, I go 5 meters north. Now I go 7 meters northwest. Then I go 7 meters west.

My clue says to look up when I see a pile of rocks. And here it is, my cache. Excellent!

And now I'll put it back for the next person. Let's take a look at where I went, graphically speaking. If I want to draw a two-dimensional representation

of where I went, we can do that on a coordinate plane, which is also called a Cartesian plane,

or a Cartesian coordinate system. We can draw left and right, up and down, using the X-axis and the Y-axis.

If we make the X-axis the east-west directions, and the Y-axis the north-south directions, we can use it to figure out the angle of the vector,

which will tell us the direction. We'll still use the tip-to-tail method, but we'll add an additional step at the end

to get our resultant. So we can draw them like this. We'll make 5 meters to the north vector a.

We label vectors with an arrow, like this, over the letter. 7 meters, 45 degrees to the northwest

is vector b. And then, 7 meters to the west is vector c. See that the three vectors are arranged tip-to-tail.

Now we can draw our resultant, using a dotted line, which is the line that connects the tail of the first vector

to the tip of the last vector, like this. To measure the magnitude of the resultant, let's use a ruler.

The resultant is 15.55 meters. When you use a protractor, you find the angle of the resultant.

And the angle is 140.2 degrees from east in a northwesterly direction.

One cool thing about graphical resolution of vectors is that we can move the vectors around and put them in a different order,

and still get the same resultant, as long as we keep them in the same direction. So we can see graphically,

that I walked 15.55 meters to find that cache. What's cool about this is that if I want to get back to my starting point,

my origin, but save myself some steps, I can use the resultant to see that I can just walk in the opposite direction.

I don't have to retrace the steps of each vector. Handy, huh? And that's a look at how you graphically resolve vectors.

Whether you're on train, geocaching, or anywhere else. And that's it for this segment of "Physics in Motion."

We'll see you next time. (announcer) For more practice problems, lab activities, and note-taking guides,

check out the "Physics in Motion" Toolkit.