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

Planes in Space

z x y Equation of a Plane

z x y

z x y

z x y

z x y b

z x y c

z x y

z x y P n

z x y Q P n r r Q Q(x,y,z) P(x 0,y 0,z 0 ) n=ai+bj+ck r=(x-x 0 )i+(y-y 0 )j+(z-z 0 )k Equation of a Plane

z x y Q P n r r Q Q(x,y,z) P(x 0,y 0,z 0 ) n=ai+bj+ck r=(x-x 0 )i+(y-y 0 )j+(z-z 0 )k Scalar Equation Vector Equation Equation of a Plane

z x y Q P n r r Q Q(x,y,z) P(x 0,y 0,z 0 ) n=ai+bj+ck r=(x-x 0 )i+(y-y 0 )j+(z-z 0 )k Scalar Equation Vector Equation Equation of a Plane

z x y Q P n r r Q Q(x,y,z) P(x 0,y 0,z 0 ) n=ai+bj+ck r=(x-x 0 )i+(y-y 0 )j+(z-z 0 )k Scalar Equation Vector Equation Equation of a Plane

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.

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

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

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

Representations of a Plane Parametric Equation Scalar Equation

Applications

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

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.

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

Distance from a Point to a Line Let P 0 be a point on l and let d be a direction vector for l. With P 0 and Q as shown in the figure, you can see that

Application

The Distance from a Point to a Plane

Application

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