Presentation on theme: "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."— 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)
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.
Find a vector equation for the line that is parallel to the vector and passes through the point
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=, in the following forms: a)Vector function b)Parametric c)Symmetric d)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 = 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 = 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=
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 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