Question

What is the dot product exactly? Like If I drew two vectors, how would I draw their dot product?

Answer 1

product for ?,??

Answer 2

The dot product isn't actually anything you can graph. It is a scalar, meaning that it is just a number (like 2 or 8 or -3.5) rather than a vector. The dot product is actually an example of something called a metric, so think of it as just another way to define a type of distance.

Answer 3

So why would people find a dot product? What does a dot product represent?

Answer 4

Dot products, like all metrics, convey a lot of information about the objects that they study. Using dot products, you can find things like the angles between vectors (like you asked in your previous problem), lengths of certain curves, areas, volumes, generalizations of volumes in higher dimensions, etc. Thus it is a very useful thing to find. As for what it represents, that is a bit harder to describe. Just try to think of it as another way to measure distance, though in a more abstract way than you would measure, say, the distance between two objects.

Answer 5

In fact, dot products and similar metrics are required to extend the ideas of calculus to objects other than just simple lines and curves. For instance, you can use a type of dot product to do calculus on spheres and other nonflat objects. The study of dot products is a very important field of mathematics, known as differential geometry, a field that is used to do things from determining the areas of mountains (nonflat objects) to studying black holes and the origins of the universe.

Answer 6

That sounds really awesome :D Thanks imperialist! BTW are you a math teacher? Because you give really cool answers.

Answer 7

Haha, I'm not actually a math teacher, I'm a junior (math major) at Duke University. But I have taught a number of classes (from middle school up to college level) and will probably eventually become a math professor, and I absolutely love helping other people learn and show them the interesting things that math can do and show. Glad you like my answers :)