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Ch 6 Analytic Geometry y x Analytic Geometry • Analytic geometry, usually called coordinate geometry or analytical geometry, is the study of geometry using the principles of algebra • The link between algebra and geometry was made possible by the development of a coordinate system which allowed geometric ideas, such as point and line, to be described in algebraic terms like real numbers and equations. • Central idea of analytic geometry – relate geometric points to real numbers. Dimensions Affine space a 0 b V – Vector space 1D – nonempty set of points y 2D x By defining each point with a z unique set of real numbers, geometric figures such as lines, circles, and conics can be 3D y described with algebraic equations. x 1 Affine space "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps“ Marcel Berger V – Vector space – nonempty set of points PPV v B A There exists point O, such that P V: a A O is one-to-onecorrespondence Affine subspace Let us consider an affine space A and its associated vector space V. Affine subspaces of A are the subsets of A of the form O V O w; w W where O is a point of A, and V a linear subspace of W. The linear subspace associated with an affine subspace is often called its direction, and two subspaces that share the same direction are said parallel. 2 Vectors in R2 • Magnitude of the vector is equal to the distance of head and tail points. A = (1,3) B = (3,1) u = A Radius vector v = B w2 = B – A w = u + v dotprod = u*v mu = Length[u] Magnitude of the cross product (with sgn) crossprod = u ⊗ w Vectors in R3 Linear combination, linear dependence Vector subspace A = (1,3,2) B = (3,1,0) O = (0,0,0) u = A v = B w = u + 2v a = Plane[A,B,O] b = PerpendicularLine[O,a] C = Point[b] n = C dotprod = n*u 3 Linear combination Let V is a vector space over a field R. x x,,,, x x x Vector x V is given by ordered tuple. 1 2 3 n Vector x is a linear combination of set ( e1, … , en ), iff there exists n-tuple ( α1, … , αn ) real numbers which yields n x ai e i a1 e 1 a 2 e 2 a n e n i1 Affine combination Take an arbitrary point A in affine space (O,V) n A O x O ai e i; where e i E i O i1 AOaEOaEO 1 1 2 2 aEOnn n A a1 E 1 a 2 E 2 ... an E n O 1 a i i1 Affine combination of points O, E1,…, En Arbitrary point A in affine space (O,V) could be expressed as an affine combination n A a1 E 1 a 2 E 2 ... an E n O 1 a i i1 n A a1 E 1 a 2 E 2 ... an E n a 0 O , where a i 1 i0 Convex combination of points O, E1,…, En - linear combination of points where all coefficients are non- negative and sum to 1 n AaEaE1 1 2 2 ... aEaOn n 0 , where a i 0 and a i 1 i0 4 Convex combination of points O, E1,…, En 1 4 AaEaEaEaEaO 1 1 2 2 3 3 4 4 0 aii, where a 0 i0 Straight line in two-dimensional space A straight line is unambiquously determined by two different points. A straight line can be analytically expressed in – Slope form y mx q – Parametric form X A t u – General equation ax by cz d 0 5 Parametric equations of a line • All points X = A + t.u where t R form a line and vice versa -- all points on that line have the form X = A + t.u for some real number t. • u – direction vector p X3 u X 2 X1 = A + u X2 = A + 2.u X1 X3 = A + 3.u X4 X4 = A + 1/2 . u A X5 = A + (-1) . u X6 X6 = A + (- 3/2) . u X5 X p X A t u ; t R Task Příklad Determine the parametric form for the line AB. u = B – A C = A+t*u GeoGebra-primka.ggb 6 Linear function y = mx + q The slope of a line m = rise over run. Calculating Slope • Slope (m) = rise (change in y) / run (change in x) • Rise is the vertical change and run is the horizontal change M = y/x M = 3/3 Rise (3) M = 1 The slope is 1. This means that for Run every increase of 1 on the x axis, (3) there will be an increase of 1 on the y axis. 7 Parametric form for the plane in 3D space X2 X3 3v 2u + 3v C v X A u B 2u 1 X1 = A + 2u u = B – A X2 = A + 3v v = C – A X3 = A + 2u + 3v X = A+t*u+s*v Parametric form for the plane in 3D space • All points X = A + t.u + s·v, where t, s R form a plane and vice versa -- all points on that plane have the form X = A + t.u + s·v for some real numbers t,s. • u, v – direction vectors X3 v A u X X A t u sv; t, s R Vektorový součin 8 General equation of the hyperplane in 2D space X p ax by c 0 p X[x; y] (X P) n(a;b) X Arbitrary point on the line p P[ p ; p ] 1 2 n Perpendicular (normal) vector of p X p X P n (for X≠P) X Pn 0 x p1 a y p2 b 0 ax by ap1 bp2 0 label: ap1 bp2 c General equation of the hyperplane in 3D space n(a,b,c) X ax by cz d 0 X[x, y, z] (X P) P[ p , p , p ] 1 2 3 Perpendicular (normal) vector of p X X P n (for X≠P) X Pn 0 x p1 a y p2 b z p3 c 0 ax by cz ap1 bp2 cp3 0 label: ap1 bp2 cp3 d 9 Conic Sections Where do you see conics in real life? 10 Circles A circle is a set of points in a plane that are equidistant from a fixed point. The distance is called the radius of the circle, and the fixed point is called the center. • A circle with center (h, k) and radius r has length r (x h)2 (y k ) 2 to some point (x, y) on the circle. • Squaring both sides yields the center-radius form of the equation of a circle. r 2 (x h)2 ( y k)2 Center-Radius Form of the Equation of a Circle The center-radius form of the equation of a circle with center (h, k) and radius r is ()()x h22 y k r 2. Notice that a circle is the graph of a relation that is not a function, since it does not pass the vertical line test. 11 Finding the Equation of a Circle Example Find the center-radius form of the equation of a circle with radius 6 and center (–3, 4). Graph the circle and give the domain and range of the relation. Solution Substitute h = –3, k = 4, and r = 6 into the equation of a circle. 62 (x (3))2 (y 4)2 36 (x 3)2 (y 4)2 General Form of the Equation of a Circle For real numbers c, d, and e, the equation x2 y2 cx dy e 0 can have a graph that is a circle, a point, or is empty. 12 Parametric equations for the circle xy221 Parametric equations for the circle x rcos t y rsin t ; t 0,2 x2 y 2 r 2 r2cos 2 t r 2 sin 2 t r 2 13 Parabola http://tube.geogebra.org/ Equations and Graphs of Parabolas A parabola is a set of points in a plane equidistant from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line the directrix, of the parabola. • For example, let the directrix be the line y = –c and the focus be the point F with coordinates (0, c). 14 Equations and Graphs of Parabolas • To get the equation of the set of points that are the same distance from the line y = –c and the point (0, c), choose a point P(x, y) on the parabola. The distance from the focus, F, to P, and the point on the directrix, D, to P, must have the same length. d(P,F) d(P,D) (x 0)2 (y c)2 (x x)2 (y (c))2 x2 y2 2yc c2 y2 2yc c2 x2 y2 2yc c2 y2 2yc c2 x2 4cy Parabola with a Vertical Axis and Vertex (0, 0) The parabola with focus (0, c) and directrix y = –c has equation x2 = 4cy. The parabola has vertex (0, 0), vertical axis x = 0, and opens upward if c > 0 or downward if c < 0. • The focal chord through the focus and perpendicular to the axis of symmetry of a parabola has length |4c|. – Let y = c and solve for x. x2 4cy x2 4c2 x 2c or 2c The endpoints of the chord are ( x, c), so the length is |4c|. 15 Determining Information about Parabolas from Equations Example Find the focus, directrix, vertex, and axis of each parabola.
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