Analytic Geometry in Space

Study equations of planes and lines in three dimensions, including distances from a point to a plane and angles between geometric objects.

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BASIC DEFINITIONS
Basic Definitions
The core formulas for planes and straight lines in space — the tools used throughout the problems below.
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1Problem 1 · Line of Intersection of Two PlanesVerify that the planes α ⁣: 2x+4y−4z+12=0\alpha\colon\ 2x+4y-4z+12=0 and β ⁣: x−3y+z=0\beta\colon\ x-3y+z=0 intersect, and then write the canonical equations of their line of intersection, aa.Start →2Problem 2 · Perpendicular from a Point to a PlaneWrite the canonical equations of the line ABAB through A(1,1,0)A(1,1,0) perpendicular to the plane α ⁣: 2x+4y−4z+12=0\alpha\colon\ 2x+4y-4z+12=0.3Problem 3 · Plane Through a Point Perpendicular to Two PlanesFind the equation of the plane γ\gamma through A(1,1,0)A(1,1,0) that is perpendicular to both planes α ⁣: 2x+4y−4z+12=0\alpha\colon\ 2x+4y-4z+12=0 and β ⁣: x−3y+z=0\beta\colon\ x-3y+z=0.4Problem 4 · Plane Through a Point and the Line of Intersection of Two PlanesFind the equation of the plane δ\delta through A(1,1,0)A(1,1,0) that contains the line of intersection of the planes α ⁣: 2x+4y−4z+12=0\alpha\colon\ 2x+4y-4z+12=0 and β ⁣: x−3y+z=0\beta\colon\ x-3y+z=0.5Problem 5 · Distance from a Point to a PlaneCompute the distance dd from the point A(1,1,0)A(1,1,0) to the plane 2x+4y−4z+12=02x+4y-4z+12=0.