Final answer:
To determine linear independence in R³, calculate the determinant of a matrix formed by the vectors in question. Zero determinant indicates dependence, non-zero indicates independence. Vectors parallel to each other or mutually perpendicular provide additional context for dependency or orthogonality. Option A is correct.
Step-by-step explanation:
The question pertains to the concept of linear independence of vectors in R³. To determine whether a set of vectors is linearly independent, one approach is to form a matrix with these vectors as columns and then calculate the determinant.
If the determinant is non-zero, the vectors are linearly independent; if it is zero, they are linearly dependent. In the case where vectors are parallel to the x-axis or mutually perpendicular to each other, there are additional considerations.
If vectors are parallel and non-zero, they are linearly dependent since they can be scaled to equal each other. If vectors are mutually perpendicular, they are orthogonal, and if they are non-zero, they are also linearly independent since no vector can be written as a combination of the others.
To determine whether a set of vectors is linearly independent or dependent, we need to check if there exists a nontrivial linear combination of the vectors that equals the zero vector. If such a combination exists, the vectors are linearly dependent; otherwise, they are linearly independent.
Let's consider the vectors v₁. We can represent them as v₁ = (x₁, y₁, z₁), where x₁, y₁, and z₁ are the components of the vector. Next, we need to set the linear combination of v₁ equal to the zero vector: c₁v₁ = (0, 0, 0), where c₁ is a scalar.
This gives us the following system of equations: c₁x₁ = 0 c₁y₁ = 0 c₁z₁ = 0 To determine if the vectors are linearly independent or dependent, we need to solve this system of equations. If there is a unique solution with c₁ = 0, then the vectors are linearly independent. Otherwise, they are dependent.