10.5 lines and planes in space parametric equations for a line in space linear equation for plane in...
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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)
INTERSECTION BETWEEN 2 PLANES
Find the angle between the planes x+2y-z=0 and x-y+3z+4=0
Angle:
Line:
1a. Write an equation for the line of intersection of the planesx + y - z = 2 and 3x - 4y + 5z = 6
1b. find the angle between the planes.
2a. Write an equation for the line of intersection of the planes 5x-3y+z-10=0 and 2x+4y-x+3=0
2b. Find the angle between the planes
Example of parallel planes will be in a future slide---for those problems, we only find the distance between the planes.
Examples of intersections of planes(note: these are not scalar multiples of each other…
therefore NOT parallel!
DISTANCE BETWEEN POINTS, PLANES, AND LINES
*This formula is from your cross product and sine formula.
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: