Are 3 Vectors Coplanar? Checking the Triple Product

In summary, the conversation discusses determining if the given vectors are coplanar. The triple product is used to find the determinant of the matrix, and although it is non-zero, the answer key claims the vectors are coplanar. The speaker questions if they or the book is wrong and asks for input. It is then clarified that the vectors are indeed coplanar, as they all exist on the same plane. The speaker even creates a program to visualize the vectors. However, the other person points out that since the determinant is 6, the vectors are linearly independent and do not lie in the same plane.
  • #1
117
0
so, I'm trying to determine if the vectors
2,1,-1
1,1,0
2,-1,3

are coplanar. i take the triple product, finding the determinant of the matrix. it seems to be non-zero, but the answer key insists these are coplanar. am i wrong, or perhaps the book? any input would be appreciated!
 
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  • #2
You're right. Did you copy the problem correctly?
 
  • #3
did you check the vectors out themselves?...? they are coplanar.
 
  • #4
coplanar means that they all exist on the same plane, right? i just made a vypthon program to draw all of the vectors as they are starting from the origin. it looks to me like the definitely span a parallelpiped, but i could certainly be wrong... now I'm even more confused
 
  • #5
Well, the determinant is 6, so the three vectors are linearly independent. Since two of the vectors span a plane and the third is not a linear combination of the former two it does not lie in the same plane.
 

1. What is the triple product in vector algebra?

The triple product is a scalar value that is calculated by taking the dot product of two vectors and then taking the cross product of the result with a third vector. In mathematical notation, it is represented as (a · b) × c.

2. How do you check if three vectors are coplanar using the triple product?

To check if three vectors are coplanar, you can use the triple product. If the triple product is equal to zero, then the vectors are coplanar. If the triple product is not equal to zero, then the vectors are not coplanar.

3. Why is the triple product used to check for coplanarity?

The triple product is used to check for coplanarity because it is a quick and efficient way to determine if three vectors lie on the same plane. It takes advantage of the properties of the cross product, where the result is a vector perpendicular to the two vectors being multiplied.

4. Can the triple product be used for more than three vectors?

Yes, the triple product can be extended to include more than three vectors. For example, for four vectors a, b, c, and d, the quadruple product is given by (a · b) × (c × d). However, the same principle applies - if the result is equal to zero, then the vectors are coplanar.

5. Are there other methods to check for coplanarity besides using the triple product?

Yes, there are other methods to check for coplanarity, such as finding the equation of the plane that the three vectors lie on and verifying if a fourth vector satisfies the equation. Another method is to check if the three vectors are linearly dependent, which would indicate that they lie on the same plane.

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