For AI agents: use /llms.txt for the Nakafa content index.
Equivalent vectors can have different positions in a plane or space, but they maintain the same magnitude and direction.
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 and are equal if:
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.
Vector equality means every corresponding component is equal. Check each axis, rather than only comparing the two magnitudes.
In a two-dimensional Cartesian plane, two vectors and are equivalent if:
This equality holds exactly when and .
In three-dimensional space, vectors and are equivalent if:
This equality holds exactly when , , and .
If vector has initial point and terminal point , then the vector can be expressed as:
Two vectors and are equivalent if:
In this equation, is the initial point and is the terminal point of .
Vector equality follows from equality of every component. Reflexivity compares a vector with itself, symmetry swaps the order of two equal vectors, and transitivity connects two equalities through a shared intermediate vector.
Every vector is equivalent to itself.
You may exchange the left and right sides of a vector equality.
If vector is equivalent to vector , then vector is also equivalent to vector .
Two equalities can be connected through the same intermediate vector.
If vector is equivalent to vector and vector is equivalent to vector , then vector is equivalent to vector .
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.
Vector with and is equivalent to vector with and .
Calculate endpoint minus starting-point coordinates for both arrows:
Since , vector is equivalent to vector .
Vector with and is equivalent to vector with and .
Calculate endpoint minus starting-point coordinates for both arrows:
Since , vector is equivalent to vector .
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.
Published: . Updated: .
