Three-dimensional Cartesian coordinates: points and vectors

Content updated: October 22, 2026

A Cartesian coordinate system represents the position of a point, or the direction and length of a vector, by an ordered triple of real numbers. Distinguish signed coordinates from distances, and points from vectors.

On this page: Coordinate axes · Coordinates · Examples · Practice · Further study

1. The Cartesian coordinate system Oxyz

Let the three axes $Ox,Oy,Oz$ have a common origin $O$ and be pairwise perpendicular. Choose unit vectors $\vec i,\vec j,\vec k$ in their positive directions. Then

These vectors form an orthonormal basis: every vector has a unique expression as a sum of scalar multiples of $\vec i,\vec j,\vec k$. For cross products, we use a right-handed coordinate system, meaning that $\vec i\times\vec j=\vec k$. Pairwise perpendicularity alone does not determine handedness; the order and orientation of the axes require this additional convention.

The planes $(Oxy),(Oyz),(Oxz)$ are spanned by the corresponding pairs of axes. Their equations are $z=0$, $x=0$, and $y=0$, respectively. Coordinate axes and planes include points with negative coordinates; a diagram shows only part of space.

2. Coordinates of vectors and points

The three coefficients are unique. Indeed, taking the dot product with each unit vector gives $x=\vec u\cdot\vec i$, $y=\vec u\cdot\vec j$, and $z=\vec u\cdot\vec k$. Thus the vector projections onto the three axes are $x\vec i$, $y\vec j$, and $z\vec k$.

Vector coordinates depend on the chosen basis; point coordinates also depend on the origin $O$. Translating a vector parallel to itself does not change its coordinates. By contrast, distinct points have distinct coordinates in a fixed coordinate system.

• For $M(x;y;z)$, its projections onto $Ox,Oy,Oz$ are $(x;0;0)$, $(0;y;0)$, and $(0;0;z)$, respectively.
• Its projections onto $(Oxy),(Oyz),(Oxz)$ are $(x;y;0)$, $(0;y;z)$, and $(x;0;z)$, respectively.
• The distances from $M$ to these three planes are $|z|$, $|x|$, and $|y|$, respectively, whereas $OM=\sqrt{x^2+y^2+z^2}$.

3. Worked examples

Example 1. Combining the three vector components

The projections of $\vec u$ onto the three axes are $\vec u_x=2\vec i$, $\vec u_y=3\vec j$, and $\vec u_z=4\vec k$. Since the axes are mutually orthogonal,

Thus $\vec u=(2;3;4)$ and $\|\vec u\|=\sqrt{29}$. Do not add the three coordinates to calculate the length.

Example 2. Reading coordinates in a rectangular box

Consider the rectangular box $ABCD.A' B' C' D'$, with $A=O$, positive directions along $AB,AD,AA'$, and corresponding lengths $3,4,2$. For brevity, denote the upper vertices by $A',B',C',D'$. Let $I$ be the midpoint of $B' C'$. The vertex and midpoint coordinates are

For example, going from $A$ to $D'$ requires moving $4$ units along $Oy$ and then $2$ units along $Oz$. The point $I$ retains the $x=3$ and $z=2$ coordinates of the edge $B' C'$, and has $y=(0+4)/2=2$. These coordinates are data of the model, not measurements taken from its perspective drawing.

Example 3. Negative coordinates and projections

Let $M(-2;1;-3)$. Its projection onto $(Oxy)$ is $H(-2;1;0)$, since only the coordinate in the normal direction $Oz$ changes. Thus $MH=|-3|=3$. Its projection onto $Ox$ is $K(-2;0;0)$; the x-coordinate of $K$ is $-2$, whereas the distance is $OK=2$.

4. Practice

1. Write the coordinates of $\vec v=-\vec i+2\vec j-3\vec k$ and calculate its length.
2. Let $P(2;-1;4)$. Find its projections onto the three coordinate planes and calculate $OP$.
3. In the box of Example 2, find the coordinates of $A',B',C'$ and check those of the midpoint $B' C'$.

5. Common mistakes

• Changing the order $x,y,z$, or replacing a signed coordinate by a nonnegative distance.
• Setting the wrong coordinate to zero in a projection: projection onto $(Oxy)$ retains $x,y$ and sets $z=0$.
• Identifying a point with a vector without specifying its position vector $\overrightarrow{OM}$.
• Concluding that a system is right-handed solely because its axes are pairwise perpendicular.

Original interactive model: Open the coordinate model on GeoGebra. Change the coordinates to observe the position of the point. The formulas in this lesson do not depend on whether the external service loads.

6. Related lessons and original resources

• Vector coordinates from the initial and terminal points (English lesson)
• The coordinate formula for the dot product (English lesson)
• Projection of a point onto a plane (English lesson)

Open the original practice resource on Scribd. This lesson includes practice problems that can be studied immediately.