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4. Three-Dimensional Coordinate Systems Vectors in the Plane Vectors in Three Dimensions The Dot Product The Cross Product Equations of Lines and Planes Cylinders and Quadric Surfaces Cylindrical and Spherical Coordinates Surfaces in Space Vectors and the Geometry of Space Advanced Topics in Single-Variable Calculus and... > Vectors and the Geometry of Space Free to share, print, make copies and changes. Get yours at www.boundless.com www.www/boundless.com/calculus?campaign_content=book_988_section_19&campaign_term=Calculus&utm_campaign=powerpoint&utm_medi um=direct&utm_source=boundless

5. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

6. Key terms azimuth an arc of the horizon intercepted between the meridian of the place and a vertical circle passing through the center of any object Cartesian of or pertaining to co-ordinates based on mutually orthogonal axes commutative such that the order in which the operands are taken does not affect their image under the operation coordinate system a method of representing points in a space of given dimensions by coordinates from an origin cross product also called a vector product; results in a vector which is perpendicular to both of the vectors being multiplied and therefore normal to the plane containing them hypersurface a -dimensional surface in a space (often a Euclidean space) of dimension manifold a topological space that looks locally like the "ordinary" Euclidean space and is Hausdorff meridian an imaginary great circle on the Earth's surface, passing through the geographic poles normal a line or vector that is perpendicular to another line, surface, or plane origin the point at which the axes of a coordinate system intersect parallelogram a convex quadrilateral in which each pair of opposite edges is parallel and of equal length parametric of, relating to, or defined using parameters Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and...

7. perpendicular at or forming a right angle (to) polynomial an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power pseudovector a quantity that transforms like a vector under a proper rotation but gains an additional change of sign under an improper rotation slope also called gradient; slope or gradient of a line describes its steepness tensor a multidimensional array satisfying a certain mathematical transformation vector a directed quantity, one with both magnitude and direction; the signed difference between two points Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and...

8. Three-Dimensional Space This is a three dimensional space represented by a Cartesian coordinate system. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Coord planes color." CC BY-SA http://en.wikipedia.org/wiki/File:Coord_planes_color.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Coord_planes_color.svgView on Boundless.com Advanced Topics in Single-Variable Calculus and...

9. Spherical Coordinate System The spherical system is used commonly in mathematics and physics and has variables of,, and. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "3D Spherical." Public domain http://en.wikipedia.org/wiki/File:3D_Spherical.svg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:3D_Spherical.svgView on Boundless.com Advanced Topics in Single-Variable Calculus and...

10. Cylindrical Coordinate System The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Coord system CY 1." Public domain http://en.wikipedia.org/wiki/File:Coord_system_CY_1.svg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:Coord_system_CY_1.svgView on Boundless.com Advanced Topics in Single-Variable Calculus and...

11. Normal Vector to a Plane This plane may be described parametrically as the set of all points of the form, where and range over all real numbers, and are given linearly independent vectors defining the plane, and is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors and can be visualized as vectors starting at and pointing in different directions along the plane. Note that and can be perpendicular but not parallel. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "PlaneR." Public domain http://en.wikipedia.org/wiki/File:PlaneR.jpg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:PlaneR.jpgView on Boundless.com Advanced Topics in Single-Variable Calculus and...

12. Vector in 3D Space A vector in the 3D Cartesian space, showing the position of a point represented by a black arrow.,, and are unit vectors along the -, -, and -axes, respectively. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Euclidean vector." CC BY http://en.wikipedia.org/wiki/Euclidean_vector View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Euclidean_vectorView on Boundless.com Advanced Topics in Single-Variable Calculus and...

13. Dot Product When finding the dot product geometrically, you need the magnitudes of the vectors and the angle between them. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Dot Product." Public domain http://en.wikipedia.org/wiki/File:Dot_Product.svg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:Dot_Product.svgView on Boundless.com Advanced Topics in Single-Variable Calculus and...

14. The Right Hand Rule If you use the rules shown in the figure, your thumb will be pointing in the direction of vector, the cross product of the vectors. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Right hand rule cross product." CC BY-SA http://en.wikipedia.org/wiki/File:Right_hand_rule_cross_product.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Right_hand_rule_cross_product.svgView on Boundless.com Advanced Topics in Single-Variable Calculus and...

15. Vertical Line, Graphed Vertical line, lying on the -plane (). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Linear equation." CC BY http://en.wikipedia.org/wiki/Linear_equation View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Linear_equationView on Boundless.com Advanced Topics in Single-Variable Calculus and...

16. Ellipsoid An ellipsoid given as. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Quadric." CC BY http://en.wikipedia.org/wiki/Quadric View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/QuadricView on Boundless.com Advanced Topics in Single-Variable Calculus and...

17. Cylinder A right circular cylinder with radius and height. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Cylinder (geometry)." CC BY http://en.wikipedia.org/wiki/Cylinder_%2528geometry%2529 View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Cylinder_%2528geometry%2529View on Boundless.com Advanced Topics in Single-Variable Calculus and...

18. Cylindrical Coordinate System A cylindrical coordinate system with origin, polar axis, and longitudinal axis. The dot is the point with radial distance, angular coordinate degrees, and height. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Cylindrical coordinate system." CC BY http://en.wikipedia.org/wiki/Cylindrical_coordinate_system View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Cylindrical_coordinate_systemView on Boundless.com Advanced Topics in Single-Variable Calculus and...

19. Spherical Coordinate System Spherical coordinates (,, ) as often used in mathematics: radial distance, azimuthal angle, and polar angle. The meanings of and have been swapped compared to the physics convention. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Spherical coordinate system." CC BY http://en.wikipedia.org/wiki/Spherical_coordinate_system View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/Spherical_coordinate_systemView on Boundless.com Advanced Topics in Single-Variable Calculus and...

20. A Sphere Defined Parametrically A sphere can be defined parametrically or implicitly. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Surface." CC BY http://en.wikipedia.org/wiki/Surface View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/SurfaceView on Boundless.com Advanced Topics in Single-Variable Calculus and...

21. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... If you want to express the point in the three dimensional coordinate system, you need to use A) two coordinates B) one coordinate C) four coordinates D) three coordinates

22. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... If you want to express the point in the three dimensional coordinate system, you need to use A) two coordinates B) one coordinate C) four coordinates D) three coordinates

23. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... In order to adequately describe a plane, you need A) a point only B) both a normal vector and a point C) either a normal vector or a point D) its normal vector only

24. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... In order to adequately describe a plane, you need A) a point only B) both a normal vector and a point C) either a normal vector or a point D) its normal vector only

25. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... In order to represent an Euclidean vector in the Cartesian coordinate system, you need to identify the coordinates of A) initial point only B) both initial and terminal points C) terminal point only D) either initial and terminal point

26. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... In order to represent an Euclidean vector in the Cartesian coordinate system, you need to identify the coordinates of A) initial point only B) both initial and terminal points C) terminal point only D) either initial and terminal point

27. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... The algebraic method to find the dot product employs the A) sum of the products of corresponding parameters B) sum of the sums of corresponding parameters C) product of the magnitudes of the vectors and the cosine of the angle between them D) product of the magnitudes of the vectors and the sine of the angle between them

28. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... The algebraic method to find the dot product employs the A) sum of the products of corresponding parameters B) sum of the sums of corresponding parameters C) product of the magnitudes of the vectors and the cosine of the angle between them D) product of the magnitudes of the vectors and the sine of the angle between them

29. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... The geometric method to find the cross product employs the A) magnitudes of the vectors and the cosine of the angle between them B) magnitudes of the vectors and the sine of the angle between them C) inputting the vector information into matrices and manipulating them D) sum of the products of corresponding parameters

30. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... The geometric method to find the cross product employs the A) magnitudes of the vectors and the cosine of the angle between them B) magnitudes of the vectors and the sine of the angle between them C) inputting the vector information into matrices and manipulating them D) sum of the products of corresponding parameters

31. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... A line is a A) three dimensional representation of a two dimensional geometric object B) two dimensional representation of a two dimensional geometric object C) three dimensional representation of a three dimensional geometric object D) two dimensional representation of a three dimensional geometric object

32. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... A line is a A) three dimensional representation of a two dimensional geometric object B) two dimensional representation of a two dimensional geometric object C) three dimensional representation of a three dimensional geometric object D) two dimensional representation of a three dimensional geometric object

33. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... A quadric surface is any D-dimensional hypersurface in A) (D -1)-dimensional space defined as the locus of zeros of a quadratic polynomial B) D-dimensional space defined as the locus of zeros of a quadratic polynomial C) (D + 1)-dimensional space defined as the locus of zeros of a quadratic polynomial D) D-dimensional space defined everywhere except the locus of zeros of a quadratic polynomial

34. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... A quadric surface is any D-dimensional hypersurface in A) (D -1)-dimensional space defined as the locus of zeros of a quadratic polynomial B) D-dimensional space defined as the locus of zeros of a quadratic polynomial C) (D + 1)-dimensional space defined as the locus of zeros of a quadratic polynomial D) D-dimensional space defined everywhere except the locus of zeros of a quadratic polynomial

35. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... If you have objects and phenomena with some rotational symmetry about the longitudinal axis, it is useful to use A) cylindrical coordinates B) spherical coordinates C) cartesian coordinates D) None of these answers

36. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... If you have objects and phenomena with some rotational symmetry about the longitudinal axis, it is useful to use A) cylindrical coordinates B) spherical coordinates C) cartesian coordinates D) None of these answers

37. Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and... To say that a surface is "two-dimensional" means that about each point there is A) a coordinate patch on which a two-dimensional coordinate system is defined B) a coordinate patch on which a three-dimensional coordinate system is defined C) a coordinate patch on which a two-dimensional coordinate system is not defined D) a coordinate patch on which a three-dimensional coordinate system is not defined

38. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Advanced Topics in Single-Variable Calculus and... To say that a surface is "two-dimensional" means that about each point there is A) a coordinate patch on which a two-dimensional coordinate system is defined B) a coordinate patch on which a three-dimensional coordinate system is defined C) a coordinate patch on which a two-dimensional coordinate system is not defined D) a coordinate patch on which a three-dimensional coordinate system is not defined

39. Attribution Wiktionary. "coordinate system." CC BY-SA 3.0 http://en.wiktionary.org/wiki/coordinate+systemCC BY-SA 3.0http://en.wiktionary.org/wiki/coordinate+system Wiktionary. "origin." CC BY-SA 3.0 http://en.wiktionary.org/wiki/originCC BY-SA 3.0http://en.wiktionary.org/wiki/origin Connexions. "Connexions - Content - Search." CC BY 3.0 http://cnx.org/content/search?words=normal+vector&allterms=weakAND&search=SearchCC BY 3.0 http://cnx.org/content/search?words=normal+vector&allterms=weakAND&search=Search Wikipedia. "Spherical coordinate system." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Spherical_coordinate_systemCC BY-SA 3.0http://en.wikipedia.org/wiki/Spherical_coordinate_system Wikipedia. "Cylindrical coordinates." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cylindrical_coordinatesCC BY-SA 3.0http://en.wikipedia.org/wiki/Cylindrical_coordinates Wiktionary. "normal." CC BY-SA 3.0 http://en.wiktionary.org/wiki/normalCC BY-SA 3.0http://en.wiktionary.org/wiki/normal Wiktionary. "vector." CC BY-SA 3.0 http://en.wiktionary.org/wiki/vectorCC BY-SA 3.0http://en.wiktionary.org/wiki/vector Wikipedia. "Plane (geometry)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Plane_(geometry)CC BY-SA 3.0http://en.wikipedia.org/wiki/Plane_(geometry) Wikipedia. "tensor." CC BY-SA 3.0 http://en.wikipedia.org/wiki/tensorCC BY-SA 3.0http://en.wikipedia.org/wiki/tensor Wiktionary. "pseudovector." CC BY-SA 3.0 http://en.wiktionary.org/wiki/pseudovectorCC BY-SA 3.0http://en.wiktionary.org/wiki/pseudovector Wikipedia. "Euclidean vector." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Euclidean_vectorCC BY-SA 3.0http://en.wikipedia.org/wiki/Euclidean_vector Wiktionary. "commutative." CC BY-SA 3.0 http://en.wiktionary.org/wiki/commutativeCC BY-SA 3.0http://en.wiktionary.org/wiki/commutative Wiktionary. "cross product." CC BY-SA 3.0 http://en.wiktionary.org/wiki/cross+productCC BY-SA 3.0http://en.wiktionary.org/wiki/cross+product Wikipedia. "Dot product." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Dot_productCC BY-SA 3.0http://en.wikipedia.org/wiki/Dot_product Wikipedia. "perpendicular." CC BY-SA 3.0 http://en.wikipedia.org/wiki/perpendicularCC BY-SA 3.0http://en.wikipedia.org/wiki/perpendicular Wiktionary. "parallelogram." CC BY-SA 3.0 http://en.wiktionary.org/wiki/parallelogramCC BY-SA 3.0http://en.wiktionary.org/wiki/parallelogram Wikipedia. "Cross product." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cross_productCC BY-SA 3.0http://en.wikipedia.org/wiki/Cross_product Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and...

40. Wiktionary. "slope." CC BY-SA 3.0 http://en.wiktionary.org/wiki/slopeCC BY-SA 3.0http://en.wiktionary.org/wiki/slope Wiktionary. "vector." CC BY-SA 3.0 http://en.wiktionary.org/wiki/vectorCC BY-SA 3.0http://en.wiktionary.org/wiki/vector Wikipedia. "Linear equation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Linear_equationCC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_equation Wiktionary. "hypersurface." CC BY-SA 3.0 http://en.wiktionary.org/wiki/hypersurfaceCC BY-SA 3.0http://en.wiktionary.org/wiki/hypersurface Wiktionary. "polynomial." CC BY-SA 3.0 http://en.wiktionary.org/wiki/polynomialCC BY-SA 3.0http://en.wiktionary.org/wiki/polynomial Wikipedia. "Cylinder (geometry)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cylinder_%2528geometry%2529CC BY-SA 3.0http://en.wikipedia.org/wiki/Cylinder_%2528geometry%2529 Wikipedia. "Quadric." CC BY-SA 3.0 http://en.wikipedia.org/wiki/QuadricCC BY-SA 3.0http://en.wikipedia.org/wiki/Quadric Wiktionary. "Cartesian." CC BY-SA 3.0 http://en.wiktionary.org/wiki/CartesianCC BY-SA 3.0http://en.wiktionary.org/wiki/Cartesian Wiktionary. "azimuth." CC BY-SA 3.0 http://en.wiktionary.org/wiki/azimuthCC BY-SA 3.0http://en.wiktionary.org/wiki/azimuth Wikipedia. "Cylindrical coordinate system." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cylindrical_coordinate_systemCC BY-SA 3.0http://en.wikipedia.org/wiki/Cylindrical_coordinate_system Wikipedia. "Spherical coordinate system." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Spherical_coordinate_systemCC BY-SA 3.0http://en.wikipedia.org/wiki/Spherical_coordinate_system Wiktionary. "manifold." CC BY-SA 3.0 http://en.wiktionary.org/wiki/manifoldCC BY-SA 3.0http://en.wiktionary.org/wiki/manifold Wiktionary. "parametric." CC BY-SA 3.0 http://en.wiktionary.org/wiki/parametricCC BY-SA 3.0http://en.wiktionary.org/wiki/parametric Wiktionary. "meridian." CC BY-SA 3.0 http://en.wiktionary.org/wiki/meridianCC BY-SA 3.0http://en.wiktionary.org/wiki/meridian Wikipedia. "Surface." CC BY-SA 3.0 http://en.wikipedia.org/wiki/SurfaceCC BY-SA 3.0http://en.wikipedia.org/wiki/Surface Free to share, print, make copies and changes. Get yours at www.boundless.com Advanced Topics in Single-Variable Calculus and...