1 / 15

Understanding the Dot Product in Vector Mathematics

This lecture provides an in-depth exploration of the dot product, a fundamental method for multiplying two vectors frequently used in three-dimensional problem-solving. It covers the mathematical formulation, laws of operation (commutative, scalar multiplication, and distribution), and Cartesian vector formulation. Additionally, it explains practical applications, such as determining the angle between two vectors and finding rectangular components. Real-world examples and assignments at the end of the lecture reinforce the concepts learned.

ula
Télécharger la présentation

Understanding the Dot Product in Vector Mathematics

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. CE 201 - Statics Lecture 6

  2. Contents • Dot Product

  3. A B  DOT PRODUCT Dot product is the method of multiplying two vectors and is used to solve three-dimensional problems If A and B are two vectors then, A . B = AB cos () where 0 180

  4. Laws of Operation (1) Commutative Law A . B = B . A (2) Multiplication by a scalar a  ( A . B ) = ( aA ) . B = A . ( aB ) = ( A . B )  a (3) Distribution Law A . ( B + D ) = ( A . B ) + ( A . D )

  5. Cartesian Vector Formulation i . i = 1 j . j = 1 k . k = 1 i . j = 0 i . k = 0 k . j = 0 i . i = (1)  (1) cos 0 = 1 i . k = (1)  (1) cos 90 = 0 If A and B are Cartesian vectors, then A . B = (Ax i + Ay j + Az k) . (Bx i + By j + Bz k = Ax Bx (i.i) + Ax By (i.j) + Ax Bz (i.k) Ay Bx (j.i) + Ay By (j.j) + Ay Bz (j.k) Az Bx (k.i) + Az By (k.j) + Az Bz (k.k) Doing Dot Product, we have A . B = Ax Bx + Ay By + Az Bz[scalar ( so there is no i, j, k)]

  6. Applications (1) to find the angle  formed between two vectors or two intersecting lines. If A and B are vectors, then  between their tails will be:  = cos-1 ( A . B ) / (AB) 0 180 If A . B = 0  = cos-1 (0) = 90 (A is perpendicular to B)

  7. A AI  a a u AII = A cos () u • to find the rectangular components of a vector. AII = A cos () AII = A cos  = A . u AII = A cos () u = (A . u) u AI = A – AII (since A = AII + AI) To find the magnitude of AI If cos () = A . U /A  = cos-1 ( A . u ) / A (0 180) AI = A sin () or AI = A2 – (AII)2 (Pythagorean)

  8. Examples • Examples 2.16 – 2.17 • Problem 2.113 • Problem 2.129

  9. Assignment No. 1 (Chapter 2) 1, 9, 12, 19, 23, 28, 29, 33, 36, 40, 47, 54, 58, 60, 65, 69, 73, 76, 78, 80

More Related