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 \( \vec n = \lt x_n,y_n,z_n \gt \).
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.
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.
