Equation of a Plane Through Three Points | Step-by-Step 3D Calculator

A plane in 3D space is uniquely determined by three non-collinear points \( P(x_1,y_1,z_1) \), \( Q(x_2,y_2,z_2) \), and \( R(x_3,y_3,z_3) \).

The normal vector \( \vec{n} \) to the plane is given by the cross product \( \vec{n} = \overrightarrow{PR} \times \overrightarrow{PQ} \). For any point \( M(x,y,z) \) on the plane, the dot product condition \( \vec{n} \cdot \overrightarrow{PM} = 0 \) yields the Cartesian equation \( ax + by + cz = d \).

\( \vec{n} = \langle a, b, c \rangle = \overrightarrow{PR} \times \overrightarrow{PQ} \), \( d = a x_1 + b y_1 + c z_1 \)

✨ The final equation is simplified by dividing all coefficients (a, b, c, d) by their greatest common divisor (GCD) to obtain the simplest integer form.

✧ Three Points Plane Solver ✧

Enter coordinates of points P, Q, and R

📍 Point \( P(x_1, y_1, z_1) \)

📍 Point \( Q(x_2, y_2, z_2) \)

📍 Point \( R(x_3, y_3, z_3) \)

📐 Plane Equation (Cartesian Form - Simplified)

\( -10x - 8y + 17z = 89 \)

📖 Step-by-Step Solution

STEP 1: Find vectors PR and PQ, then compute normal vector \( \vec{n} = \overrightarrow{PR} \times \overrightarrow{PQ} \)

STEP 2: Write the components of vector \( \overrightarrow{PM} \)

STEP 3: Orthogonality condition \( \vec{n} \cdot \overrightarrow{PM} = 0 \)

STEP 4: Expand, simplify, and divide by GCD to obtain simplest form \( ax + by + cz = d \)

💡 Geometric Interpretation

The cross product \( \overrightarrow{PR} \times \overrightarrow{PQ} \) produces a vector orthogonal to both PR and PQ, hence perpendicular to the plane containing P, Q, R. The dot product condition ensures every point M on the plane satisfies orthogonality with this normal vector.