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