Equivalence of Vector Statements: Proofs and Solutions

In summary, the statements 1) u = kv, 2) u x v = 0, 3) u * v = ||u|| * ||v||, and 4) ||u + v|| = ||u|| + ||v|| are all equivalent. This can be shown by proving that the truth of one statement implies the truth of the other three. It was shown that the truth of statement 1) implies the truth of the other three, but the converses have not been proven. However, it is possible to show that statement 4) implies statement 3), which in turn implies statement 1). This can be done by squaring statement 4) and then using the definition of the
  • #1
Bipolarity
776
2

Homework Statement


Proof that the following statements are all equivalent. First assume that none of the vectors are zero vectors. Then prove it in the degenerate case, where the vectors are zero vectors.

1) [itex] u = kv [/itex] where k is a scalar.
2) [itex] u \times v = 0 [/itex]
3) [itex] u \cdot v = ||u|| ||v|| [/itex]
4) [itex] ||u+v|| = ||u|| + ||v|| [/itex]

Homework Equations


The Attempt at a Solution


In order to prove this, we must show that the truth of each of these statements implies the truth of the other. I was able to show that the truth of the first statement implies the truth of the other three, but have not been able to show the converses. For example, how would I prove that (4) implies (1)? I would need to come up with some scalar k such that u = kv? But how could I generate this scalar?

Any ideas are appreciated.

BiP
 
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  • #2
You don't have to show they all imply 1 directly. What about 4 implies 3 implies 1? Start by squaring 4.
 

Related to Equivalence of Vector Statements: Proofs and Solutions

1. What is meant by the "equivalence of vector statements"?

The equivalence of vector statements refers to the idea that two different mathematical expressions can represent the same vector quantity. This means that if one expression is true, then the other expression must also be true, and vice versa. It is a way of showing that two statements are equivalent or equal to each other.

2. How can I prove the equivalence of two vector statements?

There are several methods for proving the equivalence of vector statements. One common method is to use algebraic manipulation to show that one statement can be transformed into the other. Another method is to use geometrical arguments, such as showing that two vectors have the same magnitude and direction. Additionally, the properties of vector operations, such as commutativity and associativity, can also be used to prove equivalence.

3. Why is it important to prove the equivalence of vector statements?

Proving the equivalence of vector statements is important because it allows us to simplify and manipulate mathematical expressions without changing their meaning. It also helps us to better understand the relationships between different vector quantities and to make connections between seemingly different expressions.

4. Can vector statements be equivalent but not identical?

Yes, vector statements can be equivalent but not identical. This means that they may have different mathematical forms, but they represent the same vector quantity. For example, the vector statement v = 2x + y may be equivalent to 2v = 4x + 2y, but they are not identical expressions.

5. How can I use the equivalence of vector statements in real-world applications?

The equivalence of vector statements is used in various fields of science and engineering, such as physics, mechanics, and computer graphics. It allows us to simplify complex equations and make calculations more efficient. It also helps in making connections between different physical quantities and understanding their relationships in real-world scenarios.

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