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

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Presentation transcript:

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

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB For vectors u and v in ℝ n 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: INNER PRODUCT

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB For a vector v in ℝ n we can define the LENGTH (or NORM) of v. If v is a vector in ℝ 2 you should recognize this as the length of the hypotenuse of a right triangle (i.e. the Pythagorean Theorem) LENGTH of a vector 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 A vector with length=1 is called a UNIT VECTOR. Any vector can be turned into a unit vector by dividing by its length. LENGTH of a vector 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 ℝ 2.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB We say that vectors u and v in ℝ n 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 uv=0 ORTHOGONALITY Here is a formula for the angle between two vectors:

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Given these vectors in ℝ 3, find the following: EXAMPLES

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Given these vectors in ℝ 3, find the following: EXAMPLES 1) Here is the calculation:

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Given these vectors in ℝ 3, find the following: EXAMPLES 2) Here is the calculation:

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

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Given these vectors in ℝ 3, find the following: EXAMPLES 4) Here is the calculation:

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Given these vectors in ℝ 3, find the following: EXAMPLES 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.