1. Vectors in R², R³ (and Rⁿ more generally).
   We start discussion in these spaces because they can be visualized. (We'll extend the notion later.) Vectors are objects in our space which have two unique properties: magnitude and direction.
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   1. Notation
   2. Direction
   3. Magnitude
   4. Location
   5. Generalizing to Rⁿ
2. Operations on vectors
   There are 2 operations which exist for all vectors in Rⁿ. They are known as addition and scalar multiplication.
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   1. Addition of Vectors Algebraically
   2. Addition of Vectors Geometrically
   3. Scalar Multiplication of Vectors Algebraically
   4. Scalar Multiplication of Vectors Geometrically
   5. Differences of Vectors Algebraically
   6. Differences of Vectors Geometrically
3. The Dot Product of Two Vectors
   The dot product of two vectors within the same space is a binary operation on vectors (ie: it requires two vectors) but instead of returning a vector, it returns a real number. The definition is simple: dotproductdefinition here
   A primary use of the dot product is the ability to determine angle measures and to determine when two vectors are perpendicular. At this level, because we may wish to extend the notion of perpendicularity to many dimensions, the word "perpendicular" and "perpendicularity" is generally replaced by the words "orthogonal" and "orthogonality".
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   1. Properties of Dot Products
   2. Angle Measure
   3. Orthogonality

4. Exercises

   1. Level 1
   2. Level 2
   3. Level 3

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