Zero Vector

High School

Definition

Also known as a null vector. A vector that has the length of 0. This means that all its components are also equal to 0. Since it doesn't have a length, its magnitude is not pointing in any direction and therefore has an undefined direction.

Worked examples

\(\vec{0} = \langle 0, 0 \rangle \) in 2D, \( \vec{0} = \langle 0, 0, 0 \rangle \) in 3D\(\)

Every component is zero, so the vector has magnitude zero and no direction.

\(\vec{v} + (-\vec{v}) = \langle 3, -2 \rangle + \langle -3, 2 \rangle = \langle 0, 0 \rangle = \vec{0}\)

Any vector plus its opposite gives the zero vector.

Common mistakes

\(\vec{0}\) has direction along the x-axis → \(\vec{0}\) has undefined direction Because the zero vector has no length, it does not point in any direction.
\(|\vec{0}| = \)undefined\(\) → \(|\vec{0}| = 0\) The magnitude is defined and equals zero; only the direction is undefined.
\(\vec{0} = 0\) → \(\vec{0} \ne 0\) The zero vector is a vector with all components zero, not the scalar zero.

Where you'll use it next

The zero vector appears when solving vector equations, finding equilibrium in physics, and studying vector spaces in linear algebra where it serves as the additive identity.

Found in 1 StudyPug lesson

Mastering Vectors: From Basics to Advanced Applications

Grade 12 Math

Dive into the world of vectors! Learn how to represent, manipulate, and apply vector concepts in various fields. Boost your problem-solving skills and prepare for advanced math and physics topics.

12th Grade12th