Lines in Space. z x y P Q Equation of a Line z x y r0r0 d P Q.

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Presentation transcript:

Lines in Space

z x y P Q Equation of a Line

z x y r0r0 d P Q

z x y r0r0 d r P Q Q’ Equation of a Line

z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) Vector Parameterization d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) Vector Parameterization d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) Vector Parameterization d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) Scalar Parametric Equations Vector Parameterization d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

Representations of a Line

Examples

Direction Cosines

Example

Examples 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 z x y 3 5 4