1 / 5

Understanding Vector Arithmetic and Norms in 2-D and 3-D Spaces

This section explores fundamental properties of vector arithmetic, highlighting essential theorems that simplify vector calculations in two and three-dimensional spaces. Key operations include commutativity, associativity, and the identities for addition and scalar multiplication. It also introduces the concept of the norm of a vector, which represents its length or magnitude. By applying the Pythagorean theorem, we establish the distance between points in space, illustrating how vector norms are used in practical applications. Lastly, we define unit vectors and their significance in vector analysis.

ernie
Télécharger la présentation

Understanding Vector Arithmetic and Norms in 2-D and 3-D Spaces

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 3.2 Norm of a Vector; Vector Arithmetic

  2. PROPERTIES OF VECTOR ARITHMETIC Theorem 3.2.1: If u, v, and w are vectors in 2- or 3-space and k and l are scalars, then the following relationships hold. (a) u + v = v + u (b) (u + v) + w = u + (v + w) (c) u + 0 = 0 + u = u (d) u + (−u) = 0 (e) k(lu) = (kl)u (f) k(u + v) = ku + kv (g) (k + l)u = ku + lu (h) 1u = u

  3. NORM OF A VECTOR The length of a vector u is often called the norm of u and is denoted by ||u||. By the Pythagorean Theorem, we have

  4. DISTANCE BETWEEN POINTS If P1(x1, y1, z1) and P2(x2, y2, z2) are two points in 3-space, then the distance between them is the norm of the vector . That is, A similar result holds for the distance between two points in 2-space.

  5. REMARKS ABOUT THE NORM • A vector that has norm 1 is called a unit vector. • From the definition of ku and the definition of norm, we have ||ku|| = |k| ||u||

More Related