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ANALYTICAL GEOMETRY INTHREE DIMENSIONSPRECALCULUSCHAPTER 111

This Slideshow was developed to accompany the textbook Precalculus By Richard Wright https://www.andrews.edu/ rwright/Precalculus-RLW/Text/TOC.html Some examples and diagrams are taken from the textbook.Slides created byRichard Wright, Andrews Academyrwright@andrews.edu2

11-01 3-D COORDINATE SYSTEMIN THIS SECTION, YOU WILL: PLOT A POINT IN 3-DIMENSIONS. CALCULATE 3-DIMENSIONAL DISTANCE AND MIDPOINT. FIND AND GRAPH THE EQUATION OF A SPHERE. FIND A TRACE OF A SPHERE.3

11-01 3-D COORDINATE SYSTEM Points in 3 dimensions (x, y, z) x comes out/into of paper y is left/right z is up/down Graph by moving out the x, over the y,then up the z. Graph A(5, 6, 3) Graph B(-2, -4, 0)4

11-01 3-D COORDINATE SYSTEM Distance Formula In 2-D: 𝑑 𝑥2 𝑥12 𝑦2 𝑦12 In 3-D: (just add the z) 𝑑 𝑥2 𝑥12 𝑦2 𝑦12 𝑧2 𝑧125

11-01 3-D COORDINATE SYSTEM Midpoint Formula In 2-D: 𝑀 𝑥1 𝑥2 𝑦1 𝑦2,22 In 3-D: (just add the z) 𝑀 𝑥1 𝑥2 𝑦1 𝑦2 𝑧1 𝑧2,,2226

11-01 3-D COORDINATE SYSTEM Equation of Circle (2-D) 𝑥 ℎ2 𝑦 𝑘2 𝑟2 Equation of Sphere (3-D) (just add z) 𝑥 ℎ2 𝑦 𝑘2 𝑧 𝑗2 𝑟2 Center is (h, k, j), r radius Graph by plotting the center andmoving each direction the radius Graph𝑥 22 𝑦 12 𝑧 12 16Center (2, -1, -1)𝑟 2 16 so 𝑟 47

11-01 3-D COORDINATE SYSTEM Trace (like intercepts for a sphere) Draw the xy trace for𝑥 2 2 𝑦 1 2 𝑧 12 16Since xy trace, let 𝑧 0𝑥 2 2 𝑦 1 2 1 2 16𝑥 2 2 𝑦 1 2 15Center (2, -1)𝑟 15 3.9Looks funny because a perspective drawing8

11-02 VECTORS IN SPACEIN THIS SECTION, YOU WILL: USE VECTOR OPERATIONS IN THREE DIMENSIONS. FIND THE ANGLE BETWEEN VECTORS.9

11-02 VECTORS IN SPACE Vectors in 2-D 𝑣റ 𝑣1 , 𝑣2 Vectors in 3-D (just add z) 𝑣റ 𝑣1 , 𝑣2 , 𝑣3 To find a vector from the initial point𝑝1 , 𝑝2 , 𝑝3 to the terminal point𝑞1 , 𝑞2 , 𝑞3 𝑣റ 𝑞1 𝑝1 , 𝑞2 𝑝2 , 𝑞3 𝑝3 If 𝑣റ 𝑣1 , 𝑣2 , 𝑣3 and 𝑢 𝑢1 , 𝑢2 , 𝑢3 , Addition Add corresponding elements 𝑣റ 𝑢 𝑣1 𝑢1 , 𝑣2 𝑢2 , 𝑣3 𝑢3 Scalar multiplication Distribute 𝑐𝑣റ 𝑐𝑣1 , 𝑐𝑣2 , 𝑐𝑣310

11-02 VECTORS IN SPACE If 𝑣റ 𝑣1 , 𝑣2 , 𝑣3 and 𝑢 𝑢1 , 𝑢2 , 𝑢3 , Dot Product 𝑣റ 𝑢 𝑣1 𝑢1 𝑣2 𝑢2 𝑣3 𝑢3 Magnitude 𝑣റ 𝑣12 𝑣22 𝑣32 Unit vector in the direction of 𝑣റ 𝑣𝑣11

11-02 VECTORS IN SPACE Angle between vectors 𝑢 𝑣റ 𝑢𝑣റ cos 𝜃 If 𝜃 90 (and 𝑢 𝑣റ 0) Then vectors are orthogonal If 𝑢 𝑐𝑣റ Then vectors are parallel12

11-02 VECTORS IN SPACE Let 𝑚 1, 0, 3 and 𝑛 2, 1, 4 Find unit vector in direction of 𝑚 Find 𝑚 Find 𝑚 2𝑛𝑚 𝑚12 𝑚22 𝑚3212 02 32 10𝑚1, 0, 3 𝑚10103 ,,10 10 10 10 3 10, 0,10101, 0, 3 2 2, 1, 41, 0, 3 4, 2, 8 3, 2, 513

11-02 VECTORS IN SPACE Let 𝑚 1, 0, 3 and 𝑛 2, 1, 4 Find the angle between 𝑚 and 𝑛 Find 𝑚 𝑛1, 0, 3 2, 1, 41 2 0 1 3 4 14𝑚 𝑛 𝑚 14 12 0232𝑛 cos 𝜃 2 2 12 4 14 10 21 cos 𝜃 14 cos 𝜃10 21𝜃 165.0 2 cos 𝜃14

11-02 VECTORS IN SPACE12 Are 𝑝റ 1, 5, 2 and 𝑞റ 5 , 1, 5 Parallel if 𝑝റ 𝑐 𝑞റparallel, orthogonal, or neither?Orthogonal if 𝑝റ 𝑞റ 0Not 0, so not orthogonal121, 5, 2 , 1,55121 5 1 25514 5 655121, 5, 2 𝑐 , 1,55Check x1 𝑐 1 𝑐 55Check y5 𝑐 1 𝑐 515

Check z2 𝑐 55c is always the same, so they are parallel 2 𝑐15

11-02 VECTORS IN SPACE Are 𝑃 1, 1, 3 , 𝑄 0, 4, 2 , and 𝑅 6, 13, 5 collinear?Find 𝑃𝑄 and 𝑄𝑅. If they are parallel, then they go in same direction.Since they would share a point, then they would be the same line.𝑃𝑄 0 1, 4 1 , 2 3 1, 5, 5𝑄𝑅 6 0, 13 4, 5 2 6, 9, 3These are not parallel because 𝑃𝑄 𝑐𝑄𝑅They are not going same direction, so not collinear16

11-03 CROSS PRODUCTSIN THIS SECTION, YOU WILL: EVALUATE A CROSS PRODUCT. USE A CROSS PRODUCT TO SOLVE AREA AND VOLUME PROBLEMS.17

11-03 CROSS PRODUCTS 𝑖റ is unit vector in x, 𝑗റ is unit vector in y, and 𝑘 is unit vector in z 𝑢 𝑢1 𝑖റ 𝑢2 𝑗റ 𝑢3 𝑘 and 𝑣റ 𝑣1 𝑖റ 𝑣2 𝑗റ 𝑣3 𝑘𝑖റ 𝑢 𝑣റ 𝑢1𝑣1𝑗റ𝑢2𝑣2𝑘𝑢3𝑣3 If 𝑢 2, 3, 3 and 𝑣റ 1, 2, 1 , find 𝑢 𝑣റ𝑗റ𝑖റ𝑗റ𝑘 𝑖റ𝑢 𝑣റ 2 3 3 2 31 2 1 1 2 3റ𝑖 3 𝑗റ 4𝑘 3𝑘 6റ𝑖 2 𝑗റ 3റ𝑖 𝑗റ 𝑘 3, 1, 118

11-03 CROSS PRODUCTS Properties of Cross Products 𝑢 𝑣റ 𝑤 𝑢 𝑣റ 𝑤 𝑢 𝑣റ 𝑣റ 𝑢 𝑢 𝑣റ is orthogonal to 𝑢 and 𝑣റ 𝑢 𝑣റ 𝑤 𝑢 𝑣റ 𝑢 𝑤 𝑢 𝑣റ 𝑢𝑣റ sin 𝜃 𝑐 𝑢 𝑣റ 𝑐𝑢 𝑣റ 𝑢 𝑐 𝑣റ 𝑢 𝑢 0 If 𝑢 𝑣റ 0, then 𝑢 and 𝑣റ areparallel19

11-03 CROSS PRODUCTS 𝐴 𝑏ℎ ℎ 𝑢 sin 𝜃 𝐴 𝑣റ𝑢 sin 𝜃 Area of a Parallelogram 𝑢 𝑣റ where 𝑢 and 𝑣റ representadjacent sides20

11-03 CROSS PRODUCTS Triple Scalar Product (shortcut)𝑢1 𝑢2 𝑢3 𝑢 𝑣റ 𝑤 𝑣1 𝑣2 𝑣3𝑤1 𝑤2 𝑤3 Volume of Parallelepiped (3-D parallelogram) 𝑉 𝑢 𝑣റ 𝑤 where 𝑢, 𝑣,റ and 𝑤represent adjacent edges21

11-04 LINES AND PLANES IN SPACEIN THIS SECTION, YOU WILL: WRITE AN EQUATION FOR A LINE IN THREE DIMENSIONS. WRITE AN EQUATION FOR A PLANE. FIND THE ANGLE BETWEEN TWO PLANES. GRAPH A PLANE.22

11-04 LINES AND PLANES IN SPACE Lines Line L goes through points P and Q 𝑣റ is a direction vector for L Start at P and move any distance indirection 𝑣റ to get some point Q 𝑃𝑄 𝑡𝑣റ because they are parallel 𝑥 𝑥1 , 𝑦 𝑦1 , 𝑧 𝑧1 𝑎𝑡, 𝑏𝑡, 𝑐𝑡 General form23

11-04 LINES AND PLANES IN SPACE Parametric Equations of Line Take each component of the generalform and solve for x, y, or z.𝑥 𝑎𝑡 𝑥1 𝑦 𝑏𝑡 𝑦1𝑧 𝑐𝑡 𝑧1 Symmetric Equation of Line Solve each equation in parametricequations for t 𝑥 𝑥1𝑎 𝑦 𝑦1𝑏 𝑧 𝑧1𝑐 We used these when we solved 3-Dsystems of equations and got manysolutions24

11-04 LINES AND PLANES IN SPACE Find a set of parametric equations of the line that passes through (1, 3, -2) and (4, 0, 1).Find the direction vector between those two points.𝑣റ 4 1, 0 3,1 2 3, 3, 3 𝑎, 𝑏, 𝑐Let’s call the first point 1, 3, 2 𝑥1 , 𝑦1 , 𝑧1Plug it in𝑥 𝑎𝑡 𝑥1𝑦 𝑏𝑡 𝑦1𝑧 𝑐𝑡 𝑧1𝑥 3𝑡 1𝑦 3𝑡 3𝑧 3𝑡 225

11-04 LINES AND PLANES IN SPACE Planes 𝑃𝑄 𝑛 0 because they areperpendicular Standard form 𝑎 𝑥 𝑥1 𝑏 𝑦 𝑦1 𝑐 𝑧 𝑧1 0 General form 𝑎𝑥 𝑏𝑦 𝑐𝑧 𝑑 026

11-04 LINES AND PLANES IN SPACE Find the general equation of plane passing through 𝐴 3, 2, 2 , 𝐵 1, 5, 0 , and 𝐶 1, 3, 1We need to find the normal vector to the plane.Find two vectors in the plane𝐴𝐵 1 3,5 2,0 2 2,3, 2𝐵𝐶 1 1, 3 5,1 0 0, 8, 1Find the cross product to get a perpendicular (normal) vector𝑛 𝐴𝐵 𝐵𝐶𝑗റ𝑖റ𝑗റ𝑘 𝑖റ𝑛 2 3 2 2 30 8 1 0 8 3റ𝑖 0റ𝑗 16𝑘 0𝑘 16റ𝑗 2റ𝑗 13റ𝑖 2റ𝑗 16𝑘 𝑎, 𝑏, 𝑐Fill in the general formI chose 𝐵 1, 5, 0 𝑥1 , 𝑦1 , 𝑧1𝑎 𝑥 𝑥1 𝑏 𝑦 𝑦1 𝑐 𝑧 𝑧1 0 13 𝑥 1 2 𝑦 5 16 𝑧 0 0Simplify to get general form 13𝑥 2𝑦 16𝑧 3 027

11-04 LINES AND PLANES IN SPACE Angle between two planes Find the angle between normalvectors Normal vectors are coefficients inthe equations of the plane 𝑛1 𝑛2 𝑛1𝑛2 cos 𝜃28

11-04 LINES AND PLANES IN SPACE Distance between a Point and a Plane 𝐷 𝑝𝑟𝑜𝑗𝑛 𝑃𝑄 𝐷 𝑃𝑄 𝑛𝑛29

11-04 LINES AND PLANES IN SPACE Graphing planes in space Find the intercepts Plot the intercepts Draw a triangle to represent theplane Sketch 3𝑥 4𝑦 6𝑧 24x-int 3𝑥 24 𝑥 8y-int 4𝑦 24 𝑦 6z-int 6𝑧 24 𝑧 430

ANALYTICAL GEOMETRY IN . Some examples and diagrams are taken from the textbook. Slides created by Richard Wright, Andrews Academy rwright@andrews.edu 2. 11-01 3-D COORDINATE SYSTEM IN THIS SECTION, YOU WILL: PLOT A POINT IN 3-DIM