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.
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