Orthonormal Vectors: Solve Your Homework

[g.lemaitre]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't understand how they get v_i * v_i = 4
What is v_i?
I can't do any work on the problem because I can't even figure out step 1.

 

[ehild]

See example 1 in your book. The base vectors v1, v2, v3, v4 are given there.

According to the further text, they should be v1=(1,1,1,1), v2=(-1,-1,1,1,), v3=(-1,1,-1,1), v4=(-1,1,1,-1).

ehild

 

[g.lemaitre]

I'm guess that v_i = v1 + v2 + v3 + v4 because that = 2 and 2*2 = 4

Is that correct?

 

[ehild]

vi means the i-th base vector. vi˙vi is the dot product of vector vi by itself.


ehild

 

[g.lemaitre]

I don't know what the i-th base vector is

 

[Fredrik]

I don't know what the i-th base vector is

For example, if a basis consists of the vectors $u_1,u_2,u_3,u_4$, and i=2, then the ith basis vector is $u_2$. Similarly, $v_i$ denotes one of the vectors $v_1,v_2,v_3,v_4$. The value of i determines which one.

 

[g.lemaitre]

For example, if a basis consists of the vectors $u_1,u_2,u_3,u_4$, and i=2, then the ith basis vector is $u_2$. Similarly, $v_i$ denotes one of the vectors $v_1,v_2,v_3,v_4$. The value of i determines which one.

the book never tells me what the value of i is.

I looked at example 1 and there is no help there.

 

[ehild]

v_i is any of the vectors v₁,v₂,v₃,v₄.

ehild

 

[g.lemaitre]

well if you take the dot product of v1 and v2 you get 0 not 4

 

[ehild]

well if you take the dot product of v1 and v2 you get 0 not 4

Who said that v1˙v2 should be 4?

ehild

 

[g.lemaitre]

It says vi * vi = 4

So if vi is any of the vectors 1 through 4, then you should be able to choose any two vectors, take the dot product and it should = 4.

In short, I want to know why vi * vi = 4

 

[HallsofIvy]

It says vi * vi = 4

So if vi is any of the vectors 1 through 4, then you should be able to choose any two vectors, take the dot product and it should = 4.

In short, I want to know why vi * vi = 4

NOT 'any two vectors'.The two 'i's are the same. vi*vi is the dot product of any one vector with itself. If they had meant the dot product of any two vectors they would have written vi*vj.

 

[g.lemaitre]

thanks, i got it now.

 

[Fredrik]

the book never tells me what the value of i is.

This author has chosen not to write out any "for all" statements. This is strictly speaking an abuse of mathematical language, but it's very common, so you will have to get used to it.

You need to be aware that in mathematical proofs, every variable that isn't assigned a specific value must be part of a "for all" or "there exists" statement. For example, these statements are all OK:

1. Since x=2, we have x²=4.
2. There exists a real number x such that x²=2.
3. For all real numbers x, we have x²≥0.

But this one isn't (strictly speaking):

x²≥0.

So when you see a statement like $v_i\cdot v_i=4$, you have to figure out if the value of i has been specified elsewhere, or if the statement is part of a "there exists" or "for all" statement. This is always easy, because an assignment of a value to a variable is never left out deliberately, and the phrase "there exists" is also never left out deliberately. So unless there's a typo, there's a "for all" missing. (That's why it can be left out without confusing people with more mathematical maturity)

In this case, it's clear that the author meant that "For all i in the set {1,2,3,4}, we have $v_i\cdot v_i=4$". And this obviously means that

$v_1\cdot v_1=4$
$v_2\cdot v_2=4$
$v_3\cdot v_3=4$
$v_4\cdot v_4=4$