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Zero Vector — Definition, Formula & Examples

Zero Vector

A vector with magnitude zero.

Key Formula

0⃗=⟨0,0,…,0⟩\vec{0} = \langle 0, 0, \ldots, 0 \rangle
Where:
  • 0⃗\vec{0} = The zero vector, with every component equal to zero

Worked Example

Problem: Given the vector v⃗=⟨3,−2⟩\vec{v} = \langle 3, -2 \rangle, find the vector u⃗\vec{u} such that v⃗+u⃗=0⃗\vec{v} + \vec{u} = \vec{0}.
Step 1: The zero vector in two dimensions is the vector with both components equal to zero.
0⃗=⟨0,0⟩\vec{0} = \langle 0, 0 \rangle
Step 2: Set up the component equations from v⃗+u⃗=0⃗\vec{v} + \vec{u} = \vec{0}.
⟨3+u1,  −2+u2⟩=⟨0,0⟩\langle 3 + u_1,\; -2 + u_2 \rangle = \langle 0, 0 \rangle
Step 3: Solve each component: u1=−3u_1 = -3 and u2=2u_2 = 2.
u⃗=⟨−3,2⟩\vec{u} = \langle -3, 2 \rangle
Answer: u⃗=⟨−3,2⟩\vec{u} = \langle -3, 2 \rangle. This is the additive inverse of v⃗\vec{v}, and their sum produces the zero vector ⟨0,0⟩\langle 0, 0 \rangle.

Why It Matters

The zero vector serves as the additive identity for vector addition: adding it to any vector v⃗\vec{v} leaves v⃗\vec{v} unchanged. This role is essential in linear algebra, where the zero vector is always included in every vector space and subspace. In physics, a zero resultant vector means forces are balanced and an object is in equilibrium.

Common Mistakes

Mistake: Saying the zero vector has a direction, or that it "points nowhere" is problematic.
Correction: The zero vector is the only vector with no defined direction. Its magnitude is zero, so direction is undefined — not "north," "up," or any other direction.

Related Terms