We will discuss vectors, which for our purposes right now are thought of as a directed line segment. Its length corresponds to its magnitude, and it encodes information regarding orientation. So we can represent, e.g., “The wind blowing 30 miles per hour due East” as a vector pointing East whose magnitude is 30.
We use vectors to discuss angles, and projections.
1. The 3 Dimensional Coordinate System. The , , axes are perpendicular to each other. We will doodle three dimensions as follows:
The formula for distance between two points and would be
A sphere with its center at would be the points which are a distance away from the center. So we would have
Usually the formula is given as
describes a sphere.
2. Vectors. A “Vector” is a directed line segment.
We know a directed line segment has both length (magnitude) and direction, so any two directed line segments with the same length and direction represent the same vector.
Vectors are “transportable” in the sense that we may translate their base point. We will represent the length of a vector as or .
The notation for a vector would be (in two dimensions) or (in three dimensions). The vector from to is given as . For and , then the vector from to is denoted .
We can add vectors graphically:
Subtraction would amount to , and graphically this is:
This describes vector addition and subtraction on the components.
Note in two-dimensional space, the vectors
are unit vectors (i.e., vectors whose length is ). They are also called basis vectors since any other vector in two-dimensions can be written as
where and are called the vector’s components. Note that the components of the vector depends on a choice of coordinates (i.e., a choice of basis vectors).
The last notion we will discuss: given any vector which is nonzero, then we can construct the unit vector
which has magnitude 1. We use hats to indicate unit vectors, and arrows for arbitrary vectors.
3. Caution: Everything stated about vectors is a half-truth. Really, these are “tangent vectors” which has a base point and a vector part (i.e., where we stick the line segment, and the directed line segment itself). We can only add/subtract two tangent vectors if they have the same base point. But since we work in Euclidean space (which is flat), we can transport vectors without a problem. This is a very special situation!
Since this is never mentioned, often students become confused when they finish vector calculus and begin studying linear algebra. Linear algebra fixes a base point, and considers the collection of all vectors sharing the same base point. This is the honest definition of a vector.
Remark 1. We will also use the phrase “three-space” instead of “three-dimensional space”, and “two-space” replacing “two-dimensional space”. In general -space is -dimensional Euclidean space.
4. Definition. Given two vectors
their “Dot Product” is the number
Let and be given vectors in three-space.
5. Angles. How do we find the angle between the two vectors?
We find that
How is this? Well, we should recall the law of cosines
which can be written as
Setting equals to equals gives us
Taking the arc cosine of both sides yields
A useful formula worth remembering
Example 1. Let and . What’s the angle between them?
Solution: We first find
We then compute
Thus the angle between and is
6. Vectors are perpendicular or “Orthogonal” if . Sometimes this is denoted .
Example 2 Consider
which implies and are orthogonal.
7. Consider the following diagram
We’re given vectors and in 3-space. Notice that if the angle between the vectors is acute, as doodled, then is the projection of onto .
However, if is obtuse, we doodle the situation thus:
Observe the projection of onto will not fall on . The projection of onto is syntactically
The natural question: what is the formula for projecting onto ? We have
The direction of depends on whether is acute or obtuse; we have its unit vector be
But now look, for both obtuse and acute we have
This describes the projection of onto for any :
Notice that its magnitude is .
8. Consider the same situation again. We have a vector as doodled
This vector is orthogonal to the projection of onto . For this reason, we write
What is it? Well, using basic vector arithmetic, we find
We will conclude our discussion of vectors here, but continue next time.