Position Vector: Formula & Examples

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  • 0:00 What Are Position Vectors?
  • 0:50 Mapping the Trip
  • 1:56 Using Arrows for the…
  • 3:05 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

It's very often necessary to describe the location of a point in space. Position vectors do this in a very precise way. In this lesson we look at the mathematical formula for position vectors and then you'll be able to test your knowledge with a quiz.

What Are Position Vectors?

It's a beautiful day for a coffee break with your friend. The plan is to meet at Carl's Coffee Cafe, buy the coffees, and then drink them on the deck. But what if you stop at Barbara's Bagel Bakery first for a bagel. Also, on your way there's Alice's Apple Auction where you could buy an apple. You sketch a map. Let's see, the Apple Auction is four blocks east from your house. The Bagel Bakery is two blocks north from the Apple Auction. The Coffee Cafe is two blocks west from the Bagel Bakery. What if your friend calls you and asks for your position? More precisely, your math-enthusiast friend specifically asks for your position vector, which you recall is the representation of a point within a given space relative to a point of origin. In this lesson, we'll look at how you could respond.

Mapping the Trip

How have we traveled on this expedition? Starting from Home, go four blocks east to buy that apple. When we are at the Apple Auction, our position is positive four along the x-axis. East is positive x and west is negative x. Of course, positive y will be north and negative y is south. It's customary to use unit vectors to show direction. The unit vector pointing in the positive x direction is the letter 'I' with a hat on top of it, î. When you write 4î, you are saying your position is four units to the right of Home. The measure of units could be feet, miles, or any convenient unit of measurement. In our coffee break example, one unit is the same as one block.

What's your position when you are at the Bagel Bakery? That's four units to the right and two units up. Using unit vectors, your position at the Bagel Bakery is 4î + 2ĵ. Note for the positive y-direction, we are using ĵ.

What is your position when you are at the Coffee Cafe? Probably sitting, right? Yes, but what is your position vector? It's 2î + 2ĵ.

Using Arrows for the Position Vector

We might as well use the letters A, B, and C to label the stops on your trip. Home could have the label O for 'origin.'

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