10.5 Lines and Planes in Space

Parametric Equations for a line in spaceLinear equation for plane in spaceSketching 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)

VECTOR VALUE FUNCTION, PARAMETRIC EQUATION, SYMMETRIC EQUATION, STANDARD FORM, AND GENERAL FORM

Equation of a line

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)

Solving for t

Write the line L through the point P = (2, 3, 5) and parallel to the vector v=<4, -1, 6>, in the following forms:a) Vector function b) Parametric c) Symmetricd) Find two points on L distinct from P.

This gives the symmetric equation of a line.

Substitute v into the equation for a line and reduce…

We can obtain an especially useful form of a line if we notice that

INTERSECTION BETWEEN TWO LINES

STANDARD EQUATION, GENERAL FORM, FUNCTIONAL FORM (*NOT IN BOOK)

Equation of a Plane

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.

Scenario 1: normal vector and point

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)

Standard Form or Point Normal Form

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.

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 x+2y-z=0 and x-y+3z+4=0

Angle:

Line:

(a Write an equation for the line of intersection of the planesx + 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 by3x-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 by10x+2y-2z -6=0 and 5x+y-z-1=0

PG. 759/#1-7ODD, 8, 9-13ODD, 14-19, 21, 25-33ODD, 37-51ODD, 63, 67-81 ODD

Homework: