# vector equation of a plane formula

This second form is often how we are given equations of planes. lx + my + nz = d. where l, m, n are the direction cosines of the unit vector parallel to the normal to the plane; (x,y,z) are the coordinates of the point on a plane and, ‘d’ is the distance of the plane from the origin. →r = →a +λ→b +μ→c f or some λ, μ ∈ R r → = a → + λ b → + μ c → f o r s o m e λ, μ ∈ R. This is the equation of the plane in parametric form. This is called the scalar equation of plane. A normal vector is, λ→b +μ→c, whereλ, μ ∈ R λ b → + μ c →, w h e r e λ, μ ∈ R. Thus, any point lying in the plane can be written in the form. Notice that if we are given the equation of a plane in this form we can quickly get a normal vector for the plane. Often this will be written as, \[ax + by + cz = d\] where \(d = a{x_0} + b{y_0} + c{z_0}\). The Cartesian equation of a plane in normal form is.

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