dotproduct - Dot(Scalar Inner Product Given two vectors u...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Dot (Scalar, Inner) Product Given two vectors u and v , the dot (scalar, inner) product between them is defined as: 1. Analytic Definition: Let u = (x1 , y1 , z1 ) and v = (x2 , y2 , z2 ). Then u · v = x1 x2 + y1 y2 + z1 z2 . 2. Geometric Definition u·v = u v cos θ where θ is the angle between u and v . Note: given two vectors, the dot product results in a scalar (a real number). Properties of Dot Product u·v u · (u + w ) αu · v O ·u = = = = v ·u u·v +u·w u · αv 0 Note 1: u · u = u 2 . Note 2: the angle θ between two non-zero vectors can be expressed in terms of the dot product as: cos θ = u·v , uv or θ = cos−1 u·v , uv Note 3: two (non-zero) vectors are perpendicular (or orthogonal, θ = π , cos θ = 0) if and only if 2 their dot product equals zero. Orthogonal Projection Given u = O and v , the orthogonal projection of v onto u is given by: Proju v = v ·u u 2 u= v· u u u = (v · u ) u ˆˆ u The scalar projection (or component) of v onto u is given by: Compu v = Note: v − Proju v ⊥ u , or equivalently (v − Proju v ) · u = 0 v· u u = (v · u ) ˆ ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern