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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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Equation of a line vector value function, parametric equation, symmetric equation, standard form, and general form

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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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**This gives the symmetric equation of a line.**

Solving for t This gives the symmetric equation of a line. Write the line L through the point P = (2, 3, 5) and parallel to the vector v=<4, -1, 6>, in the following forms: Vector function Parametric Symmetric Find two points on L distinct from P.

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

Substitute v into the equation for a line and reduce…

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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 Standard equation, general form, Functional form (*not in book)

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**Scenario 1: normal vector and point**

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.

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**Standard Form or Point Normal Form**

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)

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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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**Does it matter which point we use to plug into our standard equation?**

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= <1, 2, 1) and v = <3, 1, -2>

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**Intersection between 2 planes**

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**Find the angle between the planes**

Line: Find the angle between the planes x+2y-z=0 and x-y+3z+4=0

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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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Homework: Pg. 759/#1-7odd, 8, 9-13odd, 14-19, 21, 25-33odd, 37-51odd, 63, odd

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