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How do you assign a pair of coordinates to a vector?
Suppose a vector isn't sitting with its tail at the origin? What are its coordinates?
What's the geometric definition of the sum of two vectors?
How do you add vectors in terms of components?
What's the difference between two vectors?
What's the length of a vector?
What's a unit vector?
How do unit vectors relate to the unit circle? What are the components of a unit vector in terms of the circular functions?
Given a nonzero non-unit vector, what's a unit vector pointing in the same direction?
What is the dot product of two vectors? How does it relate to length?
What's the dot product of two unit vectors?
What's the dot product of non-unit vectors?
What are dot products good for?
What's the projection of one vector on another?
For those who find numbers comforting.
How can you write a vector as a sum of a vector parallel to a given vector and a vector perpendicular to the vector?
What is the cross product of two vectors?
A history of the vector concept (which goes well beyond what we've touched on here) can be found in the MacTutor History of Mathematics archive.