R3: U & Uperpendicular Vector Logic

• sjeddie
In summary, the conversation discusses the relationship between the x,z plane U and its perpendicular plane Uperpendicular in R3. The main confusion is between the concepts of union and direct sum, where the correct statement is that R3 is the direct sum of U and Uperp. The direct sum can be defined as a sum of elements from each set. The conversation concludes by providing an example of how every element in R3 can be written as a sum of an element in U and an element in Uperp.
sjeddie
Let's say we are in R3, and U is the x,z plane, i think then all of Uperpendicular should be some translation of the span of the y axis. Now, since that U and Uperpendicular together form R3, then isn't it true that all vectors in R3 should be contained in either U or Uperpendicular? But given a vector, say (1,1,2), it is in neither U nor Uperpendicular. What's wrong with my logic?

You have confused union with direct sum. What you have said is that $$\mathbb{R}^3 = U \cup U^\perp$$, which is not true, as you have noted. The correct statement is that $$\mathbb{R}^3 = U \oplus U^\perp$$ Note that the direct sum in this case can be defined as $$A \oplus B = \{ a + b | a \in A, b \in B \}$$. Try to show that every element of $$\mathbb{R}^3$$ may be written as a sum of an element in U and an element in $$U^\perp$$. The decomposition for your specific vector is (1,1,2) = (1,0,2) + (0,1,0).

I get it. (x,y,z)=(a,0,c) + (0,b,0) where (a,0,c) is in U and (0,b,0) is in Uperp. I didn't know what direct sum is, now I do! Thanks a lot rochfor1, you're awesome :)

1. What is R3: U & Uperpendicular Vector Logic?

R3: U & Uperpendicular Vector Logic is a mathematical concept that deals with three-dimensional vectors and their relationships to one another. It involves the use of unit vectors and perpendicular vectors to describe and manipulate vector quantities.

2. How is R3: U & Uperpendicular Vector Logic used in science?

R3: U & Uperpendicular Vector Logic is used in a variety of scientific fields, such as physics, engineering, and computer graphics. It is especially useful in describing and solving problems involving forces, motion, and three-dimensional space.

3. What is the difference between a unit vector and a perpendicular vector?

A unit vector is a vector with a magnitude of 1 and is typically used to represent a direction. A perpendicular vector is a vector that is perpendicular, or at a 90-degree angle, to another vector. They are often used together to describe the orientation and direction of a vector in three-dimensional space.

4. Can you give an example of how R3: U & Uperpendicular Vector Logic is used in real life?

Sure! A common example is in the design of bridges or buildings. Engineers use R3: U & Uperpendicular Vector Logic to calculate the forces and stresses on different structural components, ensuring that they can withstand the weight and forces they will experience in different directions.

5. Are there any limitations to using R3: U & Uperpendicular Vector Logic?

Like any mathematical concept, there are limitations to R3: U & Uperpendicular Vector Logic. It is most applicable to three-dimensional systems and may not be as useful in higher dimensions. Additionally, it may be more complex to use in cases where vector quantities are constantly changing, such as in fluid dynamics.

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