Inner Product, Length and Orthogonality Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

INNER PRODUCT For vectors u and v in ℝn we can define their INNER PRODUCT. This is also called the “dot product”. We have been using dot products to do matrix arithmetic, so it should be a familiar computation. Some properties of inner products: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

LENGTH of a vector For a vector v in ℝn we can define the LENGTH (or NORM) of v. If v is a vector in ℝ2 you should recognize this as the length of the hypotenuse of a right triangle (i.e. the Pythagorean Theorem) Note that if a vector is multiplied by a constant, then its length is multiplied by the same constant: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

LENGTH of a vector A vector with length=1 is called a UNIT VECTOR. Any vector can be turned into a unit vector by dividing by its length. u is a vector that points the same direction as v, but has length=1. When we create a unit vector in this way we say that vector v has been “normalized”. The DISTANCE between vectors u and v is the length of their difference: This should coincide with the usual “distance formula” that you know for finding the distance between 2 points in ℝ2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

u and v are orthogonal when u•v=0 ORTHOGONALITY We say that vectors u and v in ℝn are ORTHOGONAL if their inner product is 0. When two vectors are perpendicular (the angle between them is 90°) we can also call them “orthogonal”. Just a new word for a familiar property. u and v are orthogonal when u•v=0 Here is a formula for the angle between two vectors: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

EXAMPLES Given these vectors in ℝ3, find the following: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

EXAMPLES 1) Here is the calculation: Given these vectors in ℝ3, find the following: 1) Here is the calculation: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

EXAMPLES 2) Here is the calculation: Given these vectors in ℝ3, find the following: 2) Here is the calculation: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

EXAMPLES 3) Here is the calculation: Given these vectors in ℝ3, find the following: 3) Here is the calculation: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

EXAMPLES 4) Here is the calculation: Given these vectors in ℝ3, find the following: 4) Here is the calculation: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

EXAMPLES Given these vectors in ℝ3, find the following: 5) To get a unit vector, divide each component by the length of the vector: both of these unit vectors have length=1 (check this!) and point in the same directions as the original vectors. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB