Linearly Independent Vectors: Same Plane?

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In summary, three vectors are linearly independent if and only if they are not on the same plane, and three vectors are linearly dependent if and only if they are on the same plane. This is true in the context of vectors in V^3(O), or radius vectors. The statement "Three vectors are linearly independent iff The vectors are not on the same plane" means that if the vectors are not on the same plane, then they are linearly independent, and vice versa. This is the same as saying "if three vectors are linearly independent, then they are not on the same plane", which is the statement being discussed.
  • #1
kingyof2thejring
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Three vectors are linearly independent iff The vectors are not on the same plane. Three vectors are linearly dependent ⇔ The vectors are on the same plane. Is this true.
 
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  • #2
kingyof2thejring said:
Three vectors are linearly independent iff The vectors are not on the same plane. Three vectors are linearly dependent ⇔ The vectors are on the same plane. Is this true.

Yes, it is true, if you are talking about vectors in V^3(O), i.e. radius vectors.
 
  • #3
kingyof2thejring said:
Three vectors are linearly independent iff The vectors are not on the same plane. Three vectors are linearly dependent ⇔ The vectors are on the same plane. Is this true.
That's what "if and only if" (iff) means! "Three vectors are linearly independent iff The vectors are not on the same plane" means two things:
If the vectors are not on the same plane then the vectors are linearly independent" and "Three vectors are linearly independent only if the vectors are not on the same" which is the same as "if three vectors are linearly independent, then they are not on the same plane", exactly the statement you are asking about.
 

Related to Linearly Independent Vectors: Same Plane?

1. What does it mean for vectors to be linearly independent?

Linearly independent vectors are a set of vectors in which no vector can be written as a linear combination of the others. In other words, no vector in the set can be formed by adding or multiplying other vectors in the set.

2. How can you determine if two vectors are linearly independent?

To determine if two vectors are linearly independent, you can use the determinant method or the span method. The determinant method involves creating a matrix with the two vectors as columns and calculating the determinant. If the determinant is non-zero, the vectors are linearly independent. The span method involves checking if one of the vectors is a scalar multiple of the other.

3. Can three or more vectors be linearly independent?

Yes, three or more vectors can be linearly independent as long as no vector can be written as a linear combination of the others. This means that each vector is unique and cannot be formed by adding or multiplying other vectors in the set.

4. What is the importance of linearly independent vectors being in the same plane?

Vectors that are linearly independent and in the same plane are known as coplanar vectors. This is important because it allows us to easily visualize and understand the relationship between these vectors. It also allows us to use geometric methods, such as the cross product, to solve problems involving these vectors.

5. Can linearly independent vectors ever be parallel?

No, linearly independent vectors can never be parallel. If two vectors are parallel, it means that one vector is a scalar multiple of the other, which violates the definition of linear independence. However, linearly independent vectors can be coplanar, meaning they lie on the same plane, but they cannot be parallel.

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