1. Meaning of the Zero Vector
The zero vector is a vector whose magnitude is zero. All of its components are zero, so it does not point in any direction.
It is written as:
\[ \vec{0} = \langle 0, 0 \rangle \quad \text{or} \quad \langle 0, 0, 0 \rangle \]
The zero vector represents “no change” or “no movement.”
2. Properties of the Zero Vector
The zero vector has special properties that make it different from other vectors:
2.1. Zero Magnitude
The magnitude of the zero vector is always zero:
\[ |\vec{0}| = 0 \]
Since magnitude is zero, it has no specific direction.
2.2. Zero Vector in Addition
The zero vector acts like an identity element in vector addition:
\[ \vec{v} + \vec{0} = \vec{v} \]
Adding it to any vector keeps the vector unchanged.
2.3. Zero Vector in Scalar Multiplication
Any scalar multiplied with the zero vector still gives the zero vector:
\[ k \cdot \vec{0} = \vec{0} \]
3. Zero Vector vs Non-Zero Vector
A non-zero vector has both magnitude and direction. The zero vector has neither direction nor magnitude.
All components of a zero vector are zero, but a non-zero vector has at least one non-zero component.
3.1. Example Comparison
- Zero vector: \( \langle 0, 0, 0 \rangle \)
- Non-zero vector: \( \langle 2, 1, 0 \rangle \)
The non-zero vector points somewhere; the zero vector does not.
4. Examples of the Zero Vector
Here are simple situations where a zero vector appears naturally:
- If an object starts and ends at the same point, the displacement is the zero vector.
- If two equal and opposite forces act on a body, their resultant is the zero vector.
- In coordinates, subtracting a point from itself gives the zero vector:
\[ \vec{AA} = \langle x - x,\, y - y,\, z - z \rangle = \langle 0, 0, 0 \rangle \]