Lines in Space.

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Presentation transcript:

Lines in Space

z Equation of a Line Q P y x

z Equation of a Line Q d P r0 y x

z Equation of a Line Q’ Q d P r r0 y x

z Equation of a Line d r r0 y x Q’ P(x0,y0,z0) Q Q(x1,y1,z1) Q’(x,y,z) r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3 k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y x

Vector Parameterization Equation of a Line Q’ P(x0,y0,z0) Q Q(x1,y1,z1) d Q’(x,y,z) P r r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3 k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y Vector Parameterization x

Vector Parameterization Equation of a Line Q’ P(x0,y0,z0) Q Q(x1,y1,z1) d Q’(x,y,z) P r r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3 k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y Vector Parameterization x

Vector Parameterization Equation of a Line Q’ P(x0,y0,z0) Q Q(x1,y1,z1) d Q’(x,y,z) P r r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3 k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y Vector Parameterization x

Vector Parameterization Equation of a Line Q’ P(x0,y0,z0) Q Q(x1,y1,z1) d Q’(x,y,z) P r r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3 k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y Vector Parameterization x Scalar Parametric Equations

Representations of a Line

Examples

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.

Parametric Equation of a Plane z R P X Q y P Parametric Equation x

Parametric Equation of a Plane z R P X Q y P Parametric Equation x

Parametric Equation of a Plane z R P X Q y P Parametric Equation x

Representations of a Plane Scalar Equation Parametric Equation

Applications

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

Example

Example

Example

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

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

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

Examples Find the equation of of the line through the z Find the equation of of the line through the origin and perpendicular to the plane pictured. Find the equation of the plane perpendicular to x(t)=4-2t, y(t)= -1+t, z(t)=3 4 y 5 3 x

The Distance from a Point to a Plane

Distance from a Point to a Line Let P0 be a point on l and let d be a direction vector for l. With P0 and Q as shown in the figure, you can see that

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