- #1
physicsss
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Show that x+y and x-y are orthogonal if and only if x and y have the same norms.
Can someone get me started?
Can someone get me started?
Show x+y & x-y are Orthogonal if x & y Have Same Norms
- Thread starter physicsss
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- Orthogonal Vectors
In summary, if x+y and x-y are orthogonal, it means that their inner product is equal to zero. This is also equivalent to saying that the norms of x and y are equal. In a geometrical context, this can be seen as x, y, and x+y forming an isosceles right triangle. This result can also be proven algebraically by showing that the parallelogram with perpendicular diagonals formed by x and y is a rhombus, hence their equal norms.
- #2
HallsofIvy
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1) What does "orthogonal" mean here?
2) So if x+y and x-y are orthogonal what must be true?
3) And in order for that to be true what about x and y?
2) So if x+y and x-y are orthogonal what must be true?
3) And in order for that to be true what about x and y?
- #3
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Can you think of a nice geometrical application of this result
[tex] \left\langle x+y, x-y\right\rangle =0 \Longleftrightarrow ||x||=||y|| [/tex] ?
Daniel.
[tex] \left\langle x+y, x-y\right\rangle =0 \Longleftrightarrow ||x||=||y|| [/tex] ?
Daniel.
- #4
physicsss
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So (x+y)(x-y)=0, which can be turned into ||x||^2 = ||y||^2 take the square root of each side, I get ||x|| = ||y||.
As for dextercioby's question, if that is true, then x, y, and x+y make up an isosceles right triangle?
As for dextercioby's question, if that is true, then x, y, and x+y make up an isosceles right triangle?
- #5
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The paralelelogramme with perpendicular (onto another) diagonals is a rhombus. Therefore, the vectors have equal modulus. Actually, u've proven the reverse, viz.the geometrical result (theorem/proposition) by algebraic methods only. 
Daniel.
Daniel.
1. What does it mean for two vectors to be orthogonal?
Two vectors are orthogonal if they are perpendicular to each other, meaning that their dot product is equal to zero.
2. How do you determine if two vectors have the same norms?
To determine if two vectors have the same norms, you can calculate the length or magnitude of each vector using the Pythagorean theorem. If the lengths are equal, then the vectors have the same norms.
3. Can two vectors with different directions be orthogonal?
Yes, two vectors can be orthogonal even if they have different directions. As long as their dot product is equal to zero, they are considered orthogonal.
4. What is the significance of x+y and x-y being orthogonal?
When x and y have the same norms and x+y and x-y are orthogonal, it means that the two vectors are at right angles to each other and their combined length is equal to the difference of their lengths. This relationship is useful in many mathematical and scientific applications.
5. How is orthogonality used in real-world situations?
Orthogonality is used in many fields, such as physics, engineering, and computer graphics. It is used to represent perpendicular forces, angles, and distances, and is an important concept in linear algebra and vector calculus.
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