Equation of a Plane Through a point
and Perpendicular to a Vector

A calculator and solver to find the equation of a line, in 3D, that passes through a point and is perpendicular to a given vector. As many examples as needed may be generated interactively along with their detailed solutions.

Below is shown a plane through point \( P(x_p,y_p,z_p) \) and perpendicular (orthogonal) to vector \( \vec n = \lt x_n,y_n,z_n \gt \).

Plane through a point and perpendicular (orthogonal) to a vector

Since \( \vec {n} \) is perpendicular to the plane, any point \( M(x,y,z) \) is on the plane if the dot product of \( \vec n = \lt x_n,y_n,z_n \gt \) and vectors \( \vec {PM} = \lt x - x_p , y - y_p , z - z_p \gt \) is equal to zero.

\( \vec n \cdot \vec {PM} = \lt x_n,y_n,z_n \gt \cdot \lt x - x_p , y - y_p , z - z_p \gt = 0 \)

Step by step solution

STEP 1: Write the components of vector PM.

STEP 2: Write that the dot product of vectors n and PM is equal to zero.

STEP 3: Expand the product, simplify and write the equation in the form a x + b y + c z = d.

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