How do I prove that A^B is orthogonal to A?

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In summary, the conversation discusses how to show that A^B is orthogonal to A and how to express (A^B)^B as a linear combination of A and B in order to prove that it lies in the same plane as A and B. The conversation mentions using the dot product and the cross product to calculate these results.
  • #1
If A = (2,-2,1) and B = (2, 0, -1)

show by explicit calculation that;

i) A^B is orthogonal to A

ii) (A^B)^B lies in the same plane as A and B by expressing it as a linear combination of A and B

I'm using;
A^B = |A||B|sin θ

I know that when you do the cross product of two vectors the result will be a vector that is perpendicular to both and I can draw a diagram to demonstrate. However I can't show it by explicit calculation.

I thought maybe if I could prove that the angle between them was ∏/2...
 
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  • #2
When you write A^B = |A||B|sin θ, that only gives the magnitude of A^B. You need to include the direction information as well. In three dimensions the dual of A^B is a vector, which is given by the cross product. So, for part A, try forming the dot product of (AxB) with A and see what this gives.
 
  • #3
EmmaLemming said:
If A = (2,-2,1) and B = (2, 0, -1)

show by explicit calculation that;

i) A^B is orthogonal to A

ii) (A^B)^B lies in the same plane as A and B by expressing it as a linear combination of A and B

I'm using;
A^B = |A||B|sin θ

I know that when you do the cross product of two vectors the result will be a vector that is perpendicular to both and I can draw a diagram to demonstrate. However I can't show it by explicit calculation.

I thought maybe if I could prove that the angle between them was ∏/2...
Since you're talking about the cross product, why don't you write it as A x B?
 

1. How do I prove that AB is orthogonal to A?

The most common way to prove that AB is orthogonal to A is by using the dot product. If the dot product of AB and A is equal to zero, then they are orthogonal. This is because the dot product measures the angle between two vectors, and if the angle is 90 degrees (perpendicular), the dot product will be zero.

2. What is the definition of orthogonal vectors?

Orthogonal vectors are two vectors that are perpendicular to each other, meaning they form a 90-degree angle. This can also be seen in their dot product, which is equal to zero.

3. Can I prove orthogonality using the cross product?

Yes, the cross product can also be used to prove orthogonality. If the cross product of two vectors is equal to zero, then they are orthogonal.

4. Are there any other methods to prove that AB is orthogonal to A?

Yes, another method is to use the Gram-Schmidt process, which is a method for finding an orthogonal basis for a given vector space. This process involves finding the orthogonal projection of AB onto A and showing that it is equal to zero.

5. Can I use matrices to prove orthogonality?

Yes, matrices can also be used to prove orthogonality. If the product of two matrices is equal to the identity matrix, then the two matrices are orthogonal. Additionally, if the transpose of a matrix is equal to its inverse, then it is also orthogonal.

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