Mathwords logoMathwords

Coplanar — Definition, Meaning & Examples

Coplanar

Lying in the same plane. For example, any set of three points in space are coplanar.

Worked Example

Problem: Determine whether the points A(1, 0, 0), B(0, 1, 0), C(0, 0, 1), and D(1, 1, 0) are coplanar.
Step 1: Form three vectors from point A to the other three points.
AB⃗=B−A=(−1, 1, 0),AC⃗=C−A=(−1, 0, 1),AD⃗=D−A=(0, 1, 0)\vec{AB} = B - A = (-1,\, 1,\, 0), \quad \vec{AC} = C - A = (-1,\, 0,\, 1), \quad \vec{AD} = D - A = (0,\, 1,\, 0)
Step 2: Compute the scalar triple product of these three vectors. If it equals zero, the four points are coplanar.
AB⃗⋅(AC⃗×AD⃗)\vec{AB} \cdot (\vec{AC} \times \vec{AD})
Step 3: First find the cross product of AC and AD.
AC⃗×AD⃗=∣ijk−101010∣=(0−1)i−(0−0)j+(−1−0)k=(−1, 0, −1)\vec{AC} \times \vec{AD} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 0 & 1 \\ 0 & 1 & 0 \end{vmatrix} = (0-1)\mathbf{i} - (0-0)\mathbf{j} + (-1-0)\mathbf{k} = (-1,\, 0,\, -1)
Step 4: Now take the dot product with AB.
AB⃗⋅(−1, 0, −1)=(−1)(−1)+(1)(0)+(0)(−1)=1\vec{AB} \cdot (-1,\, 0,\, -1) = (-1)(-1) + (1)(0) + (0)(-1) = 1
Step 5: Since the scalar triple product equals 1, which is not zero, the four points are NOT coplanar. They do not all lie in a single plane.
Answer: The four points A, B, C, and D are not coplanar because their scalar triple product is 1, not 0.

Another Example

Problem: Are the points P(2, 0, 0), Q(0, 4, 0), R(0, 0, 6), and S(1, 2, 3) coplanar?
Step 1: Form vectors from P to the other points.
PQ⃗=(−2, 4, 0),PR⃗=(−2, 0, 6),PS⃗=(−1, 2, 3)\vec{PQ} = (-2,\, 4,\, 0), \quad \vec{PR} = (-2,\, 0,\, 6), \quad \vec{PS} = (-1,\, 2,\, 3)
Step 2: Compute the cross product of PR and PS.
PR⃗×PS⃗=∣ijk−206−123∣=(0−12)i−(−6+6)j+(−4−0)k=(−12, 0, −4)\vec{PR} \times \vec{PS} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -2 & 0 & 6 \\ -1 & 2 & 3 \end{vmatrix} = (0-12)\mathbf{i} - (-6+6)\mathbf{j} + (-4-0)\mathbf{k} = (-12,\, 0,\, -4)
Step 3: Dot with PQ.
PQ⃗⋅(−12, 0, −4)=(−2)(−12)+(4)(0)+(0)(−4)=24\vec{PQ} \cdot (-12,\, 0,\, -4) = (-2)(-12) + (4)(0) + (0)(-4) = 24
Step 4: The scalar triple product is 24, which is not zero, so these four points are not coplanar either.
Answer: P, Q, R, and S are not coplanar (scalar triple product = 24 ≠ 0).

Frequently Asked Questions

Are any 3 points always coplanar?
Yes. Any three points in space are always coplanar. You can always find at least one plane that passes through all three of them. (If the three points are collinear, infinitely many planes contain them.) It takes a fourth point to potentially break coplanarity.
How do you test if four points are coplanar?
Form three vectors from one point to the other three and compute their scalar triple product. If the result is zero, the four points are coplanar. A nonzero result means they span three-dimensional space and do not share a common plane.

Coplanar vs. Collinear

Collinear means points lie on the same line; coplanar means they lie in the same plane. Every set of collinear points is automatically coplanar, but coplanar points are not necessarily collinear. Collinearity is a stricter condition — it restricts points to one dimension, whereas coplanarity restricts them to two dimensions.

Why It Matters

Coplanarity is essential in geometry and physics whenever you need to know if objects share the same flat surface. In coordinate geometry, checking whether four points are coplanar tells you whether they define a solid region or lie flat. Engineers and architects use coplanarity to verify that structural joints meet in a single plane, which affects stability and load distribution.

Common Mistakes

Mistake: Assuming four or more points are always coplanar just because they are nearby in space.
Correction: Only three points are guaranteed to be coplanar. A fourth point may or may not lie in the same plane — you must verify, for instance with the scalar triple product.
Mistake: Confusing coplanar with colinear (collinear).
Correction: Collinear means on the same line; coplanar means in the same plane. All collinear points are coplanar, but coplanar points can be spread across a plane, not just along a line.

Related Terms

  • Plane — The flat surface coplanar objects share
  • Point — Basic element tested for coplanarity
  • Three Dimensions — Space where coplanarity becomes nontrivial
  • Collinear — Stricter condition: points on one line
  • Line — Two points define a line within a plane
  • Cross Product — Used in the scalar triple product test
  • Dot Product — Completes the scalar triple product calculation