# 10.5 Lines and Planes in Space

## Presentation on theme: "10.5 Lines and Planes in Space"— Presentation transcript:

10.5 Lines and Planes in Space
Parametric Equations for a line in space Linear equation for plane in space Sketching planes given equations Finding distance between points, planes, and lines in space

Sketching a plane

Use intercepts to find intersections with the coordinate axes (traces)

Equation of a line vector value function, parametric equation, symmetric equation, standard form, and general form

Scenario 1: Line through a point, parallel to a vector

A line corresponds to the endpoints of a set of 2-dimensional position vectors.

Vector-valued function

Find a vector equation for the line that is parallel to the vector <0, 1, -3> and passes through the point <3, -2, 0>

Scenario 2: Line through 2 points

This gives the parametric equation of a line.
are the direction numbers of the line

Find the parametric equation of a line through the points
(2, -1, 5) and (7, -2, 3)

This gives the symmetric equation of a line.
Solving for t This gives the symmetric equation of a line. Write the line L through the point P = (2, 3, 5) and parallel to the vector v=<4, -1, 6>, in the following forms: Vector function Parametric Symmetric Find two points on L distinct from P.

We can obtain an especially useful form of a line if we notice that
Substitute v into the equation for a line and reduce…

Intersection between two lines

Standard equation, general form, Functional form (*not in book)
Equation of a Plane Standard equation, general form, Functional form (*not in book)

Scenario 1: normal vector and point
Given any plane, there must be at least one nonzero vector n = <a, b, c> that is perpendicular to every vector v parallel to the plane.

Standard Form or Point Normal Form
By regrouping terms, you obtain the general form of the equation of a plane: ax+by+cz+d=0 (Standard form and general form are NOT unique!!!) Solving for “z” will get you the functional form. (unique)

Find the equation of the plane with normal n = <1, 2, 7> which contains the point (5, 3, 4). Write in standard, general, and functional form.

Scenario 2: Three non-collinear points

Find the equation of the plane passing through
(1, 2, 2), (4, 6, 1), and (0, 5 4) in standard and functional form. Note: using points in different order may result in a different normal and standard equation but the functional form will be the same.

Does it matter which point we use to plug into our standard equation?
Scenario 3: two lines Does it matter which point we use to plug into our standard equation?

Scenario 4: Line and a point not on line
Find the equation of the plane containing the point (1, 2, 2) and the line L(t) = (4t+8, t+7, -3t-2)

Scenario 5: Span of two non-parallel vectors
Note: If u and v are parallel to a given plane P, then the plane P is said to be spanned by u and v.

Find the equation of the plane through the point (0, 0, 0) spanned by the vectors u= <1, 2, 1) and v = <3, 1, -2>

Intersection between 2 planes

Find the angle between the planes
Line: Find the angle between the planes x+2y-z=0 and x-y+3z+4=0

(a Write an equation for the line of intersection of the planes
x + y - z = 2 and 3x - 4y + 5z = 6 (b) find the angle between the planes.

Distance between points, planes, and lines

Given line L that goes through the points (-3, 1, -4) and (4, 4, -6), find the distance d from the point P = (1, 1, 1) to the line L.

Finding the distance between 2 parallel planes Ex. From pg. 758
Find the distance between the two parallel planes given by 3x-y+2z -6=0 and 6x-2y+4z+4=0

Finding the distance between 2 parallel planes
Find the distance between the two parallel planes given by 10x+2y-2z -6=0 and 5x+y-z-1=0

Homework: Pg. 759/#1-7odd, 8, 9-13odd, 14-19, 21, 25-33odd, 37-51odd, 63, odd