Equivalent Vectors

Definition of Equivalent Vectors

Two vectors are said to be equivalent if they have the same magnitude (length) and direction. Mathematically, two vectors $\vec{a}$ and $\vec{b}$ are equivalent if their components are equal. In mathematical notation, this can be written as $\vec{a} \equiv \vec{b}$.

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 $\overrightarrow{CD}$ and $\overrightarrow{PQ}$ are said to be equivalent if:

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

Representation of Equivalent Vectors

In Component Form

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

where $a_1 = b_1$ and $a_2 = b_2$

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

where $a_1 = b_1$, $a_2 = b_2$, 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:

where $C(x_3, y_3)$ and $D(x_4, y_4)$

Properties of Equivalent Vectors

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 $\vec{c}$.

Examples of Equivalent Vectors

Example 1

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 $\overrightarrow{CD}$.

Example 2

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 $\overrightarrow{RS}$.

Applications of Equivalent Vectors

The concept of equivalent vectors is important in various applications, including:

1. In physics, for calculating displacement, velocity, and acceleration of objects
2. In navigation, for determining direction and travel distance
3. In computer graphics, for object transformation
4. In electrical engineering, for representing magnetic and electric forces