We introduce standard unit vectors in , and , and express a given vector as a linear combination of standard unit vectors.
A unit vector is a vector of length 1. A unit vector in the positive direction of a coordinate axis is called a standard unit vector. There are two standard unit vectors in . The vector is parallel the -axis, and the vector is parallel the -axis.
Vector names and are reserved for standard unit vectors in the direction of and axes, respectively. We chose to express and as column vectors, instead of row vectors, because the context in which we will encounter them in the future will require them to be column vectors. You may see them presented as row vectors in a different course.
There are three standard unit vectors in :
Every vector in and can be written as a sum of scalar multiples of , and . For example, if , then
When working with vectors in , we often use a different notation to denote the standard unit vectors.