When we say a vector is "greater than zero," in the context of discussing a vector's fundamental properties, it most commonly refers to the vector being a non-zero vector.
Understanding Non-Zero Vectors
Based on fundamental vector definitions, including the information provided:
- A non-zero vector is a vector that has a magnitude that is greater than zero.
- This means the vector has some "length" or "size" that is not zero.
- A key characteristic is that a non-zero vector cannot have all of its components equal to zero simultaneously.
- However, a non-zero vector can have a component with value zero, as long as at least one other component is non-zero.
Magnitude: The Key Factor
Unlike scalars (simple numbers), vectors have both magnitude (size) and direction. When comparing a vector to the scalar zero, the comparison is typically made based on its magnitude.
Think of it this way:
- The zero vector (often denoted as 0) is the vector with zero magnitude and no specific direction. All of its components are zero.
- Any other vector is a non-zero vector. Its magnitude is always a positive scalar value.
Therefore, stating a vector is "greater than zero" implies its magnitude, a scalar value, is strictly positive (greater than zero).
Characteristics of a Vector Greater Than Zero (A Non-Zero Vector)
Here's a quick summary:
- Magnitude > 0: The vector has a definable length or size that is a positive number.
- Not the Zero Vector: It is distinct from the vector where all components are zero.
- Components: While it cannot have all zero components, it can have one or more components that are zero.
Examples:
- In 2D:
- Vector v = (3, 0) is non-zero (magnitude = 3). It has a zero component.
- Vector w = (-1, 5) is non-zero (magnitude = √26 ≈ 5.1).
- Vector z = (0, 0) is the zero vector (magnitude = 0). This vector is not "greater than zero".
- In 3D:
- Vector u = (0, -2, 0) is non-zero (magnitude = 2). It has zero components.
- Vector p = (1, 1, 1) is non-zero (magnitude = √3 ≈ 1.73).
In summary, when a vector is described as "greater than zero," it means it is a non-zero vector, possessing a magnitude that is a positive value.
