Planes in Space.

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Presentation transcript:

Planes in Space

z Equation of a Plane y x

z Equation of a Plane y x

z Equation of a Plane y x

z Equation of a Plane y x

z Equation of a Plane y b x

z Equation of a Plane c y x

z Equation of a Plane y x

z Equation of a Plane n P y x

z Equation of a Plane y x P(x0,y0,z0) Q(x,y,z) n=ai+bj+ck r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y x

z Equation of a Plane y Vector Equation x Scalar Equation P(x0,y0,z0) Q(x,y,z) n=ai+bj+ck r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y Vector Equation x Scalar Equation

z Equation of a Plane y Vector Equation x Scalar Equation P(x0,y0,z0) Q(x,y,z) n=ai+bj+ck r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y Vector Equation x Scalar Equation

z Equation of a Plane y Vector Equation x Scalar Equation P(x0,y0,z0) Q(x,y,z) n=ai+bj+ck r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y Vector Equation x Scalar Equation

Examples Find the equation of the plane through (1,1,2), (3,2,-1) and (4,2,-1). Find the equation of the plane through (2,-1,3) and parallel to 3x – y + 4z =12.

Intercepts of a Plane Find the intercepts of the planes 2x – 3y + z = 6 4y + 2x = 8 z = 3 Sketch the planes. Find the normals to the planes.

Examples z Find the equation of the plane pictured. 4 y 5 3 x

Graphing Planes Sketch the following planes: (a) 3x - 2y + z = 6 (b) z + 2y = 4 (c) y = 2

Angle Between Planes Find the angle between the two planes 2x – 3y + 4z = 6 and x + 2y – 3z = -1