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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
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SKETCHING A PLANE
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Use intercepts to find intersections with the coordinate axes (traces)
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VECTOR VALUE FUNCTION, PARAMETRIC EQUATION, SYMMETRIC EQUATION, STANDARD FORM, AND GENERAL FORM
Equation of a line
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Scenario 1: Line through a point, parallel to a vector
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A line corresponds to the endpoints of a set of 2-dimensional position vectors.
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Vector-valued function
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Find a vector equation for the line that is parallel to the vector <0, 1, -3> and passes through the point <3, -2, 0>
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Scenario 2: Line through 2 points
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This gives the parametric equation of a line.
are the direction numbers of the line
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Find the parametric equation of a line through the points (2, -1, 5) and (7, -2, 3)
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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.
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Substitute v into the equation for a line and reduce…
We can obtain an especially useful form of a line if we notice that
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INTERSECTION BETWEEN TWO LINES
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STANDARD EQUATION, GENERAL FORM, FUNCTIONAL FORM (*NOT IN BOOK)
Equation of a Plane
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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
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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
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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.
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Scenario 2: Three non-collinear points
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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.
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Scenario 3: two lines
Does it matter which point we use to plug into our standard equation?
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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)
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INTERSECTION BETWEEN 2 PLANES
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Find the angle between the planes x+2y-z=0 and x-y+3z+4=0
Angle:
Line:
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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!
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DISTANCE BETWEEN POINTS, PLANES, AND LINES
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*This formula is from your cross product and sine formula.
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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.
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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
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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
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PG. 759/#1-7ODD, 8, 9-13ODD, 14-19, 21, 25-33ODD, 37-51ODD, 63, 67-81 ODD
Homework: