SUNY Geneseo Department of Mathematics
Wednesday, February 8
Math 223
Spring 2023
Prof. Doug Baldwin
(No.)
Based on “The Dot Product and Its Properties,” “Using the Dot Product to Find the Angle between Two Vectors,” and “Projections” in section 1.3 of the textbook.
The dot product of two vectors is a scalar.
The formula: ⟨x1,y1,z1⟩• ⟨x2,y2,z2⟩ = x1x2 + y1y2 + z1z2. This formula can also be used with vectors expressed in terms of the standard unit vectors.
Dot products follow algebra laws similar to many for numbers, e.g, they are commutative, scalar multiplication associates with dot product, etc.
Some uses of dot products:
Vectors are “orthogonal” if they are perpendicular. The dot product of 2 orthogonal vectors is always 0.
How to do projection calculations?
An example that might arise in projecting a vector in “global” coordinates onto a “right” viewing direction in a computer game: project ⟨1, 2, -1⟩ onto ⟨1, -1, 0⟩
There are 2 parts to this: finding the actual projection (a vector), and finding the length of the projection (a number). The length of the projection is what would actually be useful as a “right” coordinate for ⟨1, 2, -1⟩ in the game scenario.
To find the projection, consider the book’s formula for projections:
Where does this come from? A good way to think about that is that the formula is actually a fraction of the dot product times a unit vector in the direction of u. And that fraction turns out to be the length of v’s projection onto u, since the length of the projection is the length of v times the cosine of the angle between the vectors, and the dot product is the product of the magnitudes of the vectors times the cosine of the angle between them. Dividing the dot product by the magnitude of u thus leaves you with just the magnitude of v times the cosine of the angle, i.e., the length of the projection. That length times a unit vector in the direction of u gives you the projected vector.
Plugging ⟨1, -1, 0⟩ into the formula in place of u and ⟨1, 2, -1⟩ in place of v gives…
The other piece of information, the length of the projection, turns out to be simpler. The formula for it is just the part of the projection formula that we’ve already identified as giving the length of the projection, which in turn means that we’ve done most of the calculation for finding it as part of finding the projection:
What are direction cosines?
Definition: the cosines of the angles a vector makes with the x, y, and z axes.
The angles between the vector and the axes are called “direction angles”
Is there an example of orthogonality?
I claim that vectors ⟨1, 1, 0⟩ and ⟨1, -1, 3⟩ are orthogonal, but ⟨1, -1, 3⟩ and ⟨1, 2, -1⟩ are not. Visualizing the vectors in a 3-dimensional coordinate system suggests that there is indeed a 90 degree angle between the first two. Calculating dot products confirms that the dot product of the first pair of vectors is 0, while the dot product of the second pair is not.
But notice that the zero vector, ⟨0, 0, 0⟩ dotted with anything is 0, even though it doesn’t make sense to talk about the 0-length 0 vector being perpendicular to anything. Technically, we define “orthogonal” to mean having a 0 dot product, so “orthogonal” almost always means perpendicular, except when the 0 vector is involved.
Another kind of product that’s useful for, among other things, finding vectors orthogonal to a pair of known vectors.
Please read “The Cross Product and Its Properties,” “Determinants and the Cross Product,” and “Using the Cross Product” in section 1.4 of the textbook.