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# Prove that the norm of the cross product is the product of the norms if and only if A and B 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

# Prove some properties of two alternative definitions of the norm

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,

# Prove some properties of a new definition of the norm

Consider an alternative definition for the norm of a vector given by

1. Which properties of Theorems 12.4 and 12.5 (pages 453-454 of Apostol) are still valid with this new definition?
2. Draw a figure which shows the set of points of norm 1 with this new definition.

1. Claim. All of the properties of Theorems 12.4 and 12.5 hold with this new definition.
Proof. For Theorem 12.4(a), if then there is some so . Therefore .

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

2. We have the following figure,

# Prove some properties of an alternative definition of the norm of a vector

Consider an alternative definition for the norm of a vector given by

1. Prove that this definition satisfies all of the properties of Theorems 12.4 and 12.5 (pages 453-454 of Apostol).
2. Consider this definition in and draw the set of all points which have norm 1.
3. If we defined

which of the properties of Theorems 12.4 and 12.5 would hold?

1. Proof. For Theorem 12.4(a) we have

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

2. We have the following diagram

3. Property 12.4(a) fails since if we take then , but .
Property 12.4(b) holds since if then .
Property 12.4(c) holds since

Property 12.5 holds since

# Find a vector satisfying given relations with a given vector

Find a vector such that and for the following vectors ,

1. ;
2. ;
3. ;
4. .

1. If then letting we have

Since we have therefore,

Therefore, or .

2. If then letting we have

Since we have therefore,

Therefore, or .

3. If then letting we have

Since we have therefore,

Therefore, or .

4. If then letting we have

Since we have therefore,

(The final equalities follow from .) Therefore we have or .

# Compute the norms of given vectors

Given vectors , and in calculate the norms of the following vectors

1. ;
2. ;
3. ;
4. ;

1. We compute

2. We compute

3. We compute

4. We compute