Do the vectors (2,0,1),(-2,0,0) and (2,3,0) lie on the same plane? Explain your reasoning.
take the triple product
(2,0,1)•(-2,0,0)×(2,3,0)
| 2 0 1 |
|-2 0 0 | = -6
| 2 3 0 |
so the vectors are not coplanar.
The product represents the volume of the parallelopiped with the three vectors as edges. If they are coplanar, the volume is zero.
To determine whether the vectors (2,0,1), (-2,0,0), and (2,3,0) lie on the same plane, we can check if these vectors are linearly dependent or linearly independent.
Two vectors are linearly dependent if one can be written as a scalar multiple of the other. In other words, if one vector can be expressed as the sum or difference of the other two vectors. If they are linearly dependent, they lie on the same plane.
Let's check if one of these vectors can be expressed as a linear combination of the other two:
A = (2,0,1)
B = (-2,0,0)
C = (2,3,0)
To check if vector A can be expressed as a linear combination of vectors B and C, we can solve the equation: xB + yC = A, where x and y are scalars.
Using Gaussian elimination or any other appropriate method, we obtain the following system of equations:
-2x + 2y = 2
3y = 0
x = 1
From the second equation, y = 0. Substituting this into the first equation, we get -2x = 2, which implies x = -1. However, this contradicts the third equation which states that x = 1.
Since we cannot find a combination of vectors B and C that yields vector A, these vectors are linearly independent.
Therefore, the vectors (2,0,1), (-2,0,0), and (2,3,0) do not lie on the same plane.
To determine whether these vectors lie on the same plane, we can use the concept of linear dependence.
Step 1: Set up the matrix equation
Create a matrix A with the given vectors as its rows:
A = [2 0 1; -2 0 0; 2 3 0]
Step 2: Find the matrix's rank
The rank of a matrix is the maximum number of linearly independent row vectors it contains. If the rank is less than 3, it means the vectors lie on the same plane.
To find the rank, we can use various methods, such as row reduction or computing the determinant. Let's use the row reduction method:
Row reduce the matrix A:
R = rref(A)
After row reduction, we find:
R = [1 0 0; 0 1 0; 0 0 1]
Step 3: Analyze the result
Since the row-reduced echelon form of the matrix A has a full rank (3), it means the original vectors are linearly independent. Therefore, they do not lie on the same plane.
In summary, the vectors (2,0,1), (-2,0,0), and (2,3,0) do not lie on the same plane because they are linearly independent.