Download presentation

1
**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

2
Sketching a plane

3
**Use intercepts to find intersections with the coordinate axes (traces)**

4
Equation of a line vector value function, parametric equation, symmetric equation, standard form, and general form

5
**Scenario 1: Line through a point, parallel to a vector**

6
**A line corresponds to the endpoints of a set of 2-dimensional position vectors.**

7
**Vector-valued function**

8
Find a vector equation for the line that is parallel to the vector <0, 1, -3> and passes through the point <3, -2, 0>

9
**Scenario 2: Line through 2 points**

12
**This gives the parametric equation of a line.**

are the direction numbers of the line

13
**Find the parametric equation of a line through the points**

(2, -1, 5) and (7, -2, 3)

14
**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.

15
**We can obtain an especially useful form of a line if we notice that**

Substitute v into the equation for a line and reduce…

16
**Intersection between two lines**

18
**Standard equation, general form, Functional form (*not in book)**

Equation of a Plane Standard equation, general form, Functional form (*not in book)

19
**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.

20
**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)

21
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.

22
**Scenario 2: Three non-collinear points**

23
**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.

24
**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?

25
**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)

27
**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.

28
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>

29
**Intersection between 2 planes**

30
**Find the angle between the planes**

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

31
**(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.

32
**Distance between points, planes, and lines**

34
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.

37
**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

38
**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

39
Homework: Pg. 759/#1-7odd, 8, 9-13odd, 14-19, 21, 25-33odd, 37-51odd, 63, odd

Similar presentations

OK

1 What you will learn Vocabulary How to plot a point in 3 dimensional space How to plot a plane in 3 dimensional space How to solve a system of.

1 What you will learn Vocabulary How to plot a point in 3 dimensional space How to plot a plane in 3 dimensional space How to solve a system of.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on 2nd world war movies Ppt on current social evils of society Ppt on law against child marriage in saudi Ppt on energy conservation and management Ppt on conceptual artwork Marketing mix ppt on sony entertainment Ppt on total internal reflection Presentations ppt online shopping Ppt on review writing groups Ppt on teacher's day card