The Dot Product and Projections

The Cross Product

In the lecture, Justin Ryan defined the dot product between two vectors to be u and v to be

where is the angle between the vectors, with . This definition gives us a relationship connecting the dot product between two vectors with the angle between them. In the graphic below, I have created two unit vectors u and v (so we do not have to worry about length/magnitude), a slider allowing you to change the angle between these two vectors, and a display showing you the corresponding value of the dot product. Use the slider to investigate the relationship between the value of the dot product and the angle between the two vectors and answer the questions below.

Question 1

What can you say about the value of the dot product when the two vectors are pointed in relatively the same direction?

Question 2

What can you say about the value of the dot product when the two vectors are pointed in relatively opposite directions?

Question 3

What happens when the two vectors are pointing perpendicularly to each other? Do we have a special name for this relationship?

Projections in Two-Dimensional Space

In lecture, Justin also discussed the idea of a projections. There are two types of projections described in your book, the component projection and the vector projection. The component projection(or scalar projection) of the vector b onto the vector a is given by

and the vector projection of the vector b onto the vector a is given by

I have seen many students struggle with these ideas and so I have made a small demonstration below using the unit vectors from above to hopefully help you all understand this concept.
It is possible for us to discuss the projection of a onto b, or in terms of the graphic the projection of u onto v but I did not include it because I wanted to reduce clutter. We can of course define other types of projections. For example the orthogonal projection of v is defined by

Note that this projection is vector and not a scalar. An example of an orthogonal projection, using the same unit vectors, is given in the figure below along with the other projections we have studied previously.

Question 4

What do you notice about the relationship between the orthogonal projection of v and the vector u?

Question 5

Prove that the vector is orthogonal to a.

Projections in Higher-Dimensional Vectors Spaces

Finally, we can consider the idea of projections in higher-dimensional vector spaces. The highest dimension that we can visually investigate is the three-dimensional vectors and the following graphic allows you to do just that! Enter the components of any two three-dimensional vectors and you are set! Try it with the vectors and . Be sure to rotate the axes around and zoom in to get a good understanding of how these projections occur in space!