Standard Unit Vectors in
Standard Unit Vectors in and
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 :
A Vector as a Linear Combination of Standard Unit Vectors
Every vector in and can be written as a sum of scalar multiples of , and . For example, if , then
The expression is called a linear combination of , and .
Standard Unit Vectors in
When working with vectors in , we often use a different notation to denote the standard unit vectors.
Let denote a vector that has as the component and zeros elsewhere. In other
words,
where is in the position. We say that is a standard unit vector of .
Practice Problems
Problems prob:lincombijk1a-prob:lincombijk1c
Express each of the vectors as a linear combination of the appropriate standard unit vectors.
Problems prob:lincombijk2a-prob:lincombijk2c
Express each given vector in component form.
Is it possible to express as a linear combination of and alone, where and are in ?
Explain your reasoning.
2024-09-11 17:59:30