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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

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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 and passes through the point

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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=, 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.

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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 = 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 = 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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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.

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Find the equation of the plane through the point (0, 0, 0) spanned by the vectors u=

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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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(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.

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DISTANCE BETWEEN POINTS, PLANES, AND LINES

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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 by 3x-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 by 10x+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:

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