Vectors in space: endpoints and direction

Content updated: October 22, 2026

A vector describes a directed displacement. Arrows at different positions can represent the same vector, so distinguish the drawing from the properties of the vector.

Learning goals: read the endpoint order correctly, distinguish parallel vectors from vectors pointing in the same direction, and handle coincident endpoints.

Prerequisites: points, segments, parallel lines, and translations.

1. A directed segment

The notation $\overrightarrow{AB}$ denotes the vector represented by the directed segment with initial point $A$ and terminal point $B$. The arrow points from $A$ to $B$. When $A\ne B$, the line $AB$ is the supporting line of this representation, and the vector has length $AB$. The symbols $\vec a,\vec b,\vec u$ may be used when the endpoints need not be specified.

The vector $\overrightarrow{AA}$ is the zero vector, denoted $\vec0$. Its endpoints coincide, so its length is $0$. Do not call $AA$ a line or assign a particular supporting line, direction, or angle to $\vec0$.

In calculations we use free vectors: directed segments that are parallel, point in the same direction, and have the same length represent the same vector. Translating both endpoints by the same displacement preserves the vector; the two pairs of endpoints need not coincide.

2. Parallel vectors and their directions

Two vectors different from $\vec0$ are parallel if their supporting lines are parallel or coincide. They may then point in the same or opposite directions. Speak of “opposite directions” only after establishing parallelism; two different line directions are not opposite orientations.

For $\vec a\ne\vec0$, the condition $\vec b=k\vec a$ means that $\vec b$ lies along the direction of $\vec a$. If $k>0$, the vectors point in the same direction; if $k<0$, they point in opposite directions; if $k=0$, then $\vec b=\vec0$, and this directional classification does not apply. In linear-dependence criteria, cases involving the zero vector are handled algebraically.

In 3D Cartesian coordinates, the coordinates of $\overrightarrow{AB}$ equal the terminal-point coordinates minus the initial-point coordinates. Reversing the endpoints changes the sign of the vector but preserves its length. This is a quick check when a perspective sketch is hard to read; do not measure angles or lengths directly from the image.

3. Worked examples

Example 1. Parallel but oppositely directed

Suppose the vectors $\vec a,\vec b,\vec c$ have supporting lines $d_1,d_2,d_3$, respectively, where $d_1\parallel d_2$, the arrows $\vec a,\vec b$ point in opposite directions, and the direction of $d_3$ differs from that of the other two lines. Then $\vec a,\vec b$ are parallel but point in opposite directions. The vector $\vec c$ is not parallel to them, so $\vec c$ and $\vec a$ are not described as oppositely directed. These relations are given in the example, not obtained by measuring the image.

Example 2. Read the endpoints

Let $A(1;2;3)$ and $B(3;2;1)$. Then $\overrightarrow{AB}=(2;0;-2)$ and $\overrightarrow{BA}=(-2;0;2)=-\overrightarrow{AB}$. The vectors are different from $\vec0$, parallel, and oppositely directed.

Example 3. The same vector at different positions

Take $C(0;0;0)$ and $D(2;0;-2)$. Then $\overrightarrow{CD}=(2;0;-2)=\overrightarrow{AB}$ from Example 2, although $A\ne C$ and $B\ne D$. Translating the directed segment $AB$ by $\overrightarrow{AC}$ sends its endpoints to $C,D$, respectively.

4. Practice

1. Let $P(2;-1;0)$ and $Q(-1;1;4)$. Find $\overrightarrow{PQ}$ and $\overrightarrow{QP}$.
2. For $\vec a=(2;0;-2)$, classify the directions of $\vec b=(-1;0;1)$ and $\vec c=(0;1;0)$ relative to $\vec a$.
3. Given $M(1;2;3)$ and $\vec u=(2;-1;0)$, find $N$ such that $\overrightarrow{MN}=\vec u$. Does $\overrightarrow{MM}$ have a direction?

5. Common mistakes

Keep the initial and terminal points in the correct order when calculating coordinates. Do not identify “parallel” with “pointing in the same direction” or “equal.” When the endpoints coincide, handle the zero vector before discussing a supporting line, direction, or angle.

6. Further study and sources

Vectors in space: lengths and equality (English lesson) · Vector addition and subtraction (English lesson) · Vector coordinates from the initial and terminal points (English lesson).

Original resource on Scribd (external link).