Given vectors prove that
if and only if and are orthogonal.
Proof.
From Theorem 13.12(f) (page 483 of Apostol) we know
But if and only if and are orthogonal (from the definition of orthogonality). Thus, if and only if and are orthogonal
Given vectors prove that
if and only if and are orthogonal.
Proof.
From Theorem 13.12(f) (page 483 of Apostol) we know
But if and only if and are orthogonal (from the definition of orthogonality). Thus, if and only if and are orthogonal
Consider the following two definitions of the norm of a vector in .
Prove that we have the following inequalities for any vector ,
Give a geometric interpretation of this in the case that .
Proof. First, for the inequality on the left we have
For the inequality on the right, consider
Taking square roots of both sides this gives us the inequality
By induction we can then establish
Therefore,
Consider an alternative definition for the norm of a vector given by
For Theorem 12.4(b), if then for all . Therefore and so .
For Theorem 12.4(c), we compute
For Theorem 12.5 we have
Consider an alternative definition for the norm of a vector given by
which of the properties of Theorems 12.4 and 12.5 would hold?
if since all of the terms in the sum are greater than or equal to 0, with at least one non-zero since .
For Theorem 12.4(b) if then .
For Theorem 12.4(c) we have
For Theorem 12.5 we have
Property 12.5 holds since
Find a vector such that and for the following vectors ,
Since we have therefore,
Therefore, or .
Since we have therefore,
Therefore, or .
Since we have therefore,
Therefore, or .
Since we have therefore,
(The final equalities follow from .) Therefore we have or .
Given vectors , and in calculate the norms of the following vectors