Representing vectors: three-point, parallelogram, and parallelepiped rules

Content updated: October 22, 2026

To express a vector in terms of known vectors, choose a path from its initial point to its terminal point and add the displacements along it. Parallel edges in parallelograms and parallelepipeds allow equal vectors to be substituted.

Learning goals: apply the three basic rules correctly, express diagonals in terms of edges, and distinguish a valid identity from a unique representation in a basis.

Prerequisites: vector addition and subtraction, and the edges of parallelograms and parallelepipeds.

1. The three-point identity

For any three points $A,M,B$, we have $\overrightarrow{AM}+\overrightarrow{MB}=\overrightarrow{AB}$. The intermediate point $M$ need not lie on segment $AB$. In coordinates the identity is $(M-A)+(B-M)=B-A$, so it remains true for collinear or coincident points.

Additional intermediate points may make a calculation easier. Each vector must still retain its correct direction. Replacing $\overrightarrow{MB}$ by $\overrightarrow{BM}$ requires a sign change; the sum cannot be left unchanged.

2. Parallelograms and parallelepipeds

Let $ABCD$ be a parallelogram with its vertices listed in order around the boundary. Then $\overrightarrow{BC}=\overrightarrow{AD}$, so

Use the two adjacent edges starting at $A$. Adding one edge twice cannot replace the sum of adjacent edges: $\overrightarrow{AD}+\overrightarrow{AD}=2\overrightarrow{AD}$ is generally not the diagonal $\overrightarrow{AC}$.

For a nondegenerate parallelepiped $ABCD.A'B'C'D'$, set $\vec u=\overrightarrow{AB}$, $\vec v=\overrightarrow{AD}$, and $\vec w=\overrightarrow{AA'}$. The upper vertices are the translates of $A,B,C,D$, respectively, by $\vec w$. Then $\overrightarrow{AC'}=\vec u+\vec v+\vec w$.

Starting at $A'$, the three-point identity implies

Indeed, the three vectors on the left are $-\vec w,\vec u,\vec v$, respectively, whereas $\overrightarrow{A'C}=\vec u+\vec v-\vec w$. The edges of the parallelepiped need not be pairwise perpendicular.

The three edges of a nondegenerate parallelepiped are linearly independent, so every vector in space has a unique representation in $\vec u,\vec v,\vec w$. Two independent vectors represent only vectors in the vector plane they span. Do not omit independence and still conclude uniqueness.

3. Worked examples

Example 1. The other diagonal of a parallelogram

For $\vec u=\overrightarrow{AB}$ and $\vec v=\overrightarrow{AD}$, we have $\overrightarrow{BD}=\overrightarrow{BA}+\overrightarrow{AD}=-\vec u+\vec v$, whereas $\overrightarrow{AC}=\vec u+\vec v$. The two diagonals have different formulas.

Example 2. The midpoint of a space diagonal

Let $I$ be the midpoint of $AC'$. Then

If $J$ is the midpoint of $BD'$, then

The vectors are equal and have the same initial point $A$, so $I=J$.

Example 3. Check a coordinate parallelepiped

Take $A=O$, $\vec u=(2;0;0)$, $\vec v=(1;3;0)$, and $\vec w=(0;0;4)$. Then $C'=(3;3;4)$, $A'=(0;0;4)$, and $C=(3;3;0)$. We have $\overrightarrow{A'C}=(3;3;-4)=\vec u+\vec v-\vec w$. This is a parallelepiped but not a rectangular box, since $\vec u\cdot\vec v=2\ne0$.

4. Practice

1. In a parallelogram with $\overrightarrow{AB}=\vec u$ and $\overrightarrow{AD}=\vec v$, express $\overrightarrow{CA}$ and $\overrightarrow{DB}$.
2. In the parallelepiped above, express $\overrightarrow{B'D}$ in terms of $\vec u,\vec v,\vec w$.
3. Using the coordinate vectors in Example 3, calculate $\overrightarrow{B'D}$ and locate the center of the parallelepiped.

5. Common mistakes

Use the correct vertex order in a parallelogram and the corresponding vertex pairs of a parallelepiped. Do not infer perpendicular or equal edges solely from a perspective sketch. To conclude that two points coincide from equal vectors, those vectors must have the same initial point.

6. Further study and sources

Vector addition and subtraction (English lesson) · Coplanar vectors (English lesson) · Midpoint coordinates in space (English lesson).

Original resource on Scribd (external link).