Determine if the given vectors are orthogonal

[g.lemaitre]

Homework Statement

Homework Equations

The Attempt at a Solution

A set of vectors are orthogonal if any two are perpendicular. the cross product of w1 and w2 is

-9 + 2 + 3 + 4 = 0

So the set of vectors is orthogonal. The book says that's false. Why?

 

[ehild]

The set of vectors are orthogonal if any pair of them is orthogonal. That means not only a single pair being orthogonal, but all pairs, that is

w1*w2=w1*w3=w1*w4=w2*w3=w2*w4=w3*w4=0

w2 is orthogonal to w1, but not to w3. Calculate w2*w3, is it zero? What about w3 and w4?

ehild

 

[g.lemaitre]

Well, I find the book's use of the English language very unfortunate. If I say a group of people is multi-ethic if any two of its members are of a different race, then it is multiethic. That means only two of its members need be different, not all of them. But if orthogonal means that all pairs must perpendicular then that's the way it is.

 

[ehild]

That is the definition of an orthogonal set of vectors. All possible pairs are orthogonal. Choosing
any two vectors, they are orthogonal. The multi-ethic group means that you can choose at least one pair of people who belong to different races. I would not use "any" in this case.

ehild

 

[g.lemaitre]

Like I said, all possible is not equal to any two

 

[Mark44]

Well, I find the book's use of the English language very unfortunate. If I say a group of people is multi-ethic if any two of its members are of a different race, then it is multiethic. That means only two of its members need be different, not all of them. But if orthogonal means that all pairs must perpendicular then that's the way it is.

That is the definition of an orthogonal set of vectors. All possible pairs are orthogonal. Choosing
any two vectors, they are orthogonal. The multi-ethic group means that you can choose at least one pair of people who belong to different races. I would not use "any" in this case.

"multiethic" does not mean multiple races.

The definition you show for orthogonality is incorrect (translation error?). Here's a simple counterexample.
Let S = {<1, 0, 0>, <0, 1, 0>, <1, 1, 0>}

Clearly, the first two vectors in the list above are orthogonal, so by the posted definition, the entire set is orthogonal. However, taking dot products, we see that <1, 0, 0>$\cdot$ <1, 1, 0> = 1, so these two vectors aren't orthogonal.

Likewise, <0, 1, 0>$\cdot$ <1, 1, 0> = 1, so these two vectors aren't orthogonal, either.

For a set of vectors to be orthogonal, every pair of them must be orthogonal.

 

[Mark44]

the cross product of w1 and w2 is
-9 + 2 + 3 + 4 = 0

What you've done is the dot product, not the cross product. The result of the cross product of two vectors is another vector, not a scalar.

 

[g.lemaitre]

What you've done is the dot product, not the cross product. The result of the cross product of two vectors is another vector, not a scalar.

I often confuse the two.