Equivalent Vectors

Equal Vectors at Different Starting Points

Two vectors are equal, or represented by equivalent directed segments, when their components are equal. For nonzero vectors, this is the same as having equal magnitude and the same direction. For vectors $\vec{a}$ and $\vec{b}$, we write $\vec{a} = \vec{b}$.

Equivalent vectors can have different positions in a plane or space, but they maintain the same magnitude and direction.

Conditions for Equivalent Vectors

Two vectors are equal only when both conditions hold at the same time, so check both before you call them equal. A pair that satisfies only one of the two conditions is not equivalent.

Two nonzero vectors $\overrightarrow{CD}$ and $\overrightarrow{PQ}$ are equal if:

1. Both vectors have equal length: $|\overrightarrow{CD}| = |\overrightarrow{PQ}|$
2. Both vectors have the same direction

Representation of Equivalent Vectors

Two vectors are equivalent when they share the same length and the same direction, so either one can stand in for the other. That replacement moves a vector to a convenient starting point without changing the vector itself.

In Component Form

In a two-dimensional Cartesian plane, two vectors $\vec{a}$ and $\vec{b}$ are equivalent if:

This equality holds exactly when $a_1 = b_1$ and $a_2 = b_2$.

In three-dimensional space, vectors $\vec{a}$ and $\vec{b}$ are equivalent if:

This equality holds exactly when $a_1 = b_1$, $a_2 = b_2$, and $a_3 = b_3$.

In Terms of Initial and Terminal Points

If vector $\overrightarrow{AB}$ has initial point $A(x_1, y_1)$ and terminal point $B(x_2, y_2)$, then the vector can be expressed as:

Two vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are equivalent if:

In this equation, $C(x_3, y_3)$ is the initial point and $D(x_4, y_4)$ is the terminal point of $\overrightarrow{CD}$.

Rules That Follow from Vector Equality

Each of the three rules keeps the endpoints of a chain and removes one repeated vector, so a long list collapses into a single relation. Intermediate vectors can be left out without changing the statement.

Equality of vectors behaves like equality of numbers. The three rules below let a chain of equivalent vectors be shortened.

Reflexive Property

Every vector is equivalent to itself.

Symmetric Property

If vector $\vec{a}$ is equivalent to vector $\vec{b}$, then vector $\vec{b}$ is also equivalent to vector $\vec{a}$.

Transitive Property

If vector $\vec{a}$ is equivalent to vector $\vec{b}$ and vector $\vec{b}$ is equivalent to vector $\vec{c}$, then vector $\vec{a}$ is equivalent to vector $\vec{c}$.

Checking Equivalent Vectors from Their Components

Two vectors are equivalent when the change in the first coordinate matches and the change in the second coordinate matches. The examples below compare those changes directly.

Equality from Matching Component Changes

Vector $\overrightarrow{AB}$ with $A(2, 3)$ and $B(5, 7)$ is equivalent to vector $\overrightarrow{CD}$ with $C(1, 1)$ and $D(4, 5)$.

Proof:

Since $\overrightarrow{AB} = \overrightarrow{CD} = (3, 4)$, vector $\overrightarrow{AB}$ is equivalent to vector $\overrightarrow{CD}$.

Equivalent Vectors with Different Starting Points

Vector $\overrightarrow{PQ}$ with $P(0, 0)$ and $Q(2, 2)$ is equivalent to vector $\overrightarrow{RS}$ with $R(3, 1)$ and $S(5, 3)$.

Proof:

Since $\overrightarrow{PQ} = \overrightarrow{RS} = (2, 2)$, vector $\overrightarrow{PQ}$ is equivalent to vector $\overrightarrow{RS}$.

Why Starting Points Can Differ

Equivalent vectors have the same magnitude and direction even when their starting points differ. The same displacement can therefore be drawn at different positions without changing the vector. This is useful when comparing translations or moving a vector within a geometric construction.