Chapter 1: Vectors, Lines and Planes
Section 1.3: Dot Product
Show that A·A=0 is a necessary and sufficient condition for A to be the zero vector.
Let A be a vector whose components are the real numbers ak,k=1,…,n, and assume A·A=0.
Now A·A=∑k=1nak2, so the condition that A·A=0 means a sum of positive real numbers adds to zero, something that can only happen if each of the numbers is itself zero. Hence, all the components of A are zero, and that makes A the zero vector.
Now assume that A is the zero vector so that all its components are zero. Since A·A=∑k=1nak2 and all the ak are zero, the sum must be zero, and therefore A·A=0.
<< Previous Example Section 1.3
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)