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# 4: The Vector Space Rⁿ

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• 4.1: Vectors in Rⁿ
The notation $$\mathbb{R}^{n}$$
• 4.2: Vector Algebra
Addition and scalar multiplication are two important algebraic operations done with vectors. We will explore these operations in more detail in the following sections.
• 4.3: Geometric Meaning of Vector Addition
• 4.4: Length of a Vector
In this section, we explore what is meant by the length of a vector in $$\mathbb{R}^n$$.
• 4.5: Geometric Meaning of Scalar Multiplication
• 4.6: Parametric Lines
We can use the concept of vectors and points to find equations for arbitrary lines in $$\mathbb{R}^n$$, although in this section the focus will be on lines in $$\mathbb{R}^3$$.
• 4.7: The Dot Product
There are two ways of multiplying vectors which are of great importance in applications. The first of these is called the dot product.
• 4.8: Planes in Rⁿ
Much like the above discussion with lines, vectors can be used to determine planes in $$\mathbb{R}^n$$.
• 4.9: The Cross Product
Recall that the dot product is one of two important products for vectors. The second type of product for vectors is called the cross product.
• 4.10: Spanning, Linear Independence and Basis in Rⁿ
By generating all linear combinations of a set of vectors one can obtain various subsets of $$\mathbb{R}^{n}$$ which we call subspaces. For example what set of vectors in $$\mathbb{R}^{3}$$ generate the $$XY$$-plane? What is the smallest such set of vectors can you find? The tools of spanning, linear independence and basis are exactly what is needed to answer these and similar questions and are the focus of this section.
• 4.11: Orthogonality
In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
• 4.12: Applications
Vectors are used to model force and other physical vectors like velocity.

Thumbnail: Animation showing how the vector cross product (green) varies when the angles between the blue and red vectors is changed. (Public Domain; Nicostella via Wikipedia)

4: The Vector Space Rⁿ is shared under a CC BY license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) .