Analytic Geometry MAT 035 5.09.04 WEEKLY PROGRAM - The first week of the semester, we will introduce the course and given a brief outline. We continue with vectors in R n and some operations including addition of two vectors. The second week, we talk about inner product, vector product (cross product) and their geometric interpretations. We also give the projection of a vector along another one and perpendicular and parallel vectors. Furthermore we give some related theorems and some (un)solved examples. Furthermore, we are going to give the mixed product with their geometric interpretation. If time permits we will start the new chapter LINE. In the tutorial section, you will see points of division, and solve the following exercises. Tutorial. Let V be a vector space and <, > an inner product on V. a) Show that < 0, u >= 0 for all u in V. b) Show that < u, v >= 0 for all v in V, then u = 0.. Let <, > be standard inner product on R. [ ] [ ] a) Let α = and β =. Find a vector γ such that < α, γ >= and < β, γ >= 3. b) Show that α =< α, e > e + < α, e > e for any α in R. (Here, the vectors e and e are standard basis vectors of R.) [ ] 3 3. Find the norm of α = with respect to 4 a) the usual inner product. b) the inner product given as < x, y >= x y x y x y + 3x y. 4. Sketch a directed line segment in R 3 representing each of the following vectors:
a) A vector with tail (beginning point) A(,, 5) and head (end point) B(3,, ). b) A vector with tail (beginning point) A(,, 4) and head (end point) B(, 4, 3). 5. Find the vector AB, and their lengths, and also sketch them in R 3 if a) A(,, 0) and B(,, 7). b) A(5,, 3) and B(,, 0). 6. Determine the beginning point of the vector (,, 3). 7. For what values of a, b and c are the vectors equal? 6 0 a b a b 6 whose end point is and 8. Compute u + v, u v, u 3 v, 3 u + v and sketch them if a) u = and v = 4 3. b) u = 0 and v = 4. a + b c 9. Show that AB + BC + CA = 0 for a triangle ABC. 0. Show that AD = ( AC + AB) for the median vector AD of a triangle ABC.. In the figure, the points A, B and C are collinear. If AC = 3 AB and DC = k AD + pdb then k p =? A B C D
. Let G be a barycenter of a triangle ABC. Show that a) GA + GB + b) for any point K, GC = 0 KA + KB + KC = 3 KG. 3. Let ABC be a triangle, and D and E be the midpoints of AB and AC. Then show that DE is parallel to BC ( DE BC) and DE =. BC 3 4. If the vectors u = m and v = 3m are parallel, then find the n m 8m values of m and n. [ ] 5. Let u = and v = v ω = 3. [ ] be given. Find ω such that u ω = and 6. Find the norm of given vectors, and also find the unit vectors in directions of them. 3 a) u = 3 b) v = 4 5 7. Find the projection of the vector u = 8. Find the angle between the vectors u and v if a) u =, v =. b) u = 0, v = 0. onto v = 9. In R n if u + v = u v then find the angle between the vectors u and v. 3 4 5.
0. Let a, b and c be nonzero vectors. If a b = a c, then can we necessarily say that we have b = c?. Prove the following inequalities: a) Triangle inequality : a + b a + b for all a, b and c in R n. b) Schwartz inequality : a b a b for all a, b and c in R n.. Let a and b mutually perpendicular non-zero vectors. Show that for any number λ, a + λ b a. 3. For any vectors a, b in n-space, prove the followings: a) a + b + a b = ( a + ) b. b) a + b a b = 4 a b. 4. If possible, find a, b and c so that v = and y =. a b c is orthogonal to both x = 5. Let ABC be a triangle. Prove the cosine theorem, i.e. BC = AC + AB AC AB cosâ AC = BC + AB BC AB cos ˆB 6. Compute the followings: AB = BC + AC BC AC cos Ĉ. a) e e b) e e 3 c) e e d) e e 3
e) e 3 e f) ( e e ) e 3 g) ( e e ) e 3 h) x y if x = i) u v if u = 5 8 and y = and v = 7. If u v = u ω then can we necessarily say that v = ω? 8. Given u =, v = 3 6 and ω = compute a) ( u v) ω b) u ( v ω) c) ( u v) ω d) u ( v ω) 9. Let u, v and ω be arbitrary vectors in 3-space. Is there any difference between ( u v) ω and u ( v ω)? 30. Find a nonzero vector which is perpendicular to both u and v given as follows: a) u = and v =. 3 0 b) u = c) u = 0 0 3 and v = and v = 0. 3. 4.. 3. Let u, v and ω be mutually perpendicular vectors in R 3. Show that ( u v) ω = 0.
3. Let u, v, ω and t be arbitrary vectors in 3-space. Prove the following expansions: a) ( u v) ( ω t) = ( u, v, t) ω ( u, v, ω) t = ( u, ω, t) v ( v, ω, t) u b) ( u v) ( ω t) = ( u ω)( v t) ( v ω)( u t) 33. Find the coordinates of midpoint of the line segment joining the points P (3, 7) and P (, 3). 34. Find the coordinates of the point P which divides the line segment joining the points P (, 5) and P (4, ), in the ratio at 6 and, respectively. 5 35. Find the slope and direction cosines of the line joining the points P (, ) and P ( 5, 3). 36. Find the least angle of the triangle ABC, the coordinates of whose vertices are A(, 4), B( 5, ) and C(0, 6). 37. Find the angle between the directed lines joining P (, 3), P ( 4, 3), and P 3 (, 0) and P 4 ( 5, 6) by slopes and direction cosines. 38. Show that the two lines joining P (7, 5), P (, ), and P 3 (4, 3) and P 4 (, 0) are perpendicular. Bedia Akyar Moller
Analytic Geometry MAT 035 9.09.04 WEEKLY PROGRAM 3-4 I am going to remind you scalar product, vector product and mixed product and their geometric interpretations. Furthermore, I will give some related theorems and solve some examples. I am going to start the new chapter line. I will start with direction numbers, direction angles (direction cosines), perpendicular or parallel lines. Then I will give the equation of a straight line, the intercept form of a straight line and the normal form of the equation of a straight line in plane and the angle between two lines in terms of tangents and their direction cosines. Solve the following exercises: Tutorial. Find the slope and direction cosines of the line which is perpendicular to the line joining the points P (, 4) and P (, ).. Find the angle between the directed lines joining P (, 3), P ( 4, 3), and P 3 (, 0) and P 4 ( 5, 6) by slopes and direction cosines. 3. Given the triangle whose vertices are A( 5, 6), B(, 4) and C(3, ) derive the equations of the three medians and solve algebraically for their point of intersection. 4. Find the intercepts of the line perpendicular to x + 3y 7 = 0 and passing through the point (, 6). 5. Find the equations of the lines through (, 6) if the product of the intercepts for each line is. 6. What are the direction cosines of a line perpendicular to x 5y + 3 = 0. 7. Find the equations of all the lines with slope m = 3 4 will occur an area 4 unit with the coordinate axes. such that each line 8. Determine the parameter k such that a) the line 3kx + 5y + k = 0 passes through A(, 4).
b) the line 4x ky 7 has the slope 3. c) the line whose equation x + ky + 3 = 0 shall make an angle 45 o with the line x + 5y 7 = 0. 9. Find the perpendicular distances from origin to the lines a) x 3y + 6 = 0 b) 5x 8y 5=0 0. Find the equation of the line through the point of intersection of the lines x 3y + = 0 and x + 5y 9 = 0 and whose distance from the origin is 5.. Calculate the distance d from the line 5x y 3 = 0 to point (, 3). Are the point (, 3) and the origin on the same side?. Determine the equations of the bisectors of the angles between the lines l : 3x + 4y = 0 and l : x 5y + 5 = 0. 3. Write the equations of the line whose parametric equations are x = 5 t, y = 3 + t, z = 3t in symmetric form. 4. Write the equations of the line through the points (7,, ) and (3,, 4). 5. Write the equations of the line parallel to x = y + 4 = z 3 (4,, 3). and through 6. Discuss the intersection of the lines l and l if a) l : x + = y = z 3 and l : x + = 6 y = z +. b) l : x + = y = z 3 and l : x 4 = y = z 5. c) l : x + = y = z 3 and l : x 3 = y 4 = z. Bedia Akyar Moller
Analytic Geometry MAT 035 0.0.04 WEEKLY PROGRAM 5-6 We have started the chapter straight line and seen direction numbers, direction angles (direction cosines), perpendicular or parallel lines. We have also given the symmetric and parametric equations of a straight line, the slope of a line, the intercept form of a line, the normal form of the equation of a straight line in R, the angle between two lines, division of points. We am planning to start the new chapter plane. We will give the vector form of the equation of a plane, the angle between two planes and we will also give other forms of the equation of a plane. I will also give the distance from a point to a plane. We will mention some situations between planes and lines we will again talk about the intersections of two planes and some examples by different methods. We will see intersections of three planes and intersections of lines and planes and finally the specialized distance formulae and finish the chapter plane. If time permits, we will start the new chapter Locus Problems. We will also give circles determined various conditions. We will do some examples. I highly recommend you to read Matematik Dunyasi, 005 Yaz, Guz, Konikler. Solve the following exercises:. Show that the two lines passing through P (7, 5), P (, ) and P 3 (4, 3), P 4 (, 0) are perpendicular.. Find the area of the triangle A(, ), B(5, 3) and C( 8, 0). 3. Show that three points A(, 5), B(6, ) and C( 4, ) are colinear (that is, they lie on a line). 4. Prove that the diagonals of a rectangle are equal. 5. Prove that the diagonals of a parallelogram bisect each other. 6. Prove that the medians of a triangle intersect in a point 3 along one from a vertex toward the opposite side. of the distance 7. Find the intercepts of the line perpendicular to x + 3y 7 = 0 passing through the point (,6). 8. What are the direction cosines of a line perpendicular to x 5y + 3 = 0.
9. Find the equations of all lines with slope m = 3 such that each line will 4 occur an area 4 unit with the coordinate axes. 0. Determine the parameter k such that i) The line 3kx + 5y + k = 0 passes through A(, 4). ii) The line 4x ky 7 = 0 has the slope 3.. Write the equations of the line x = 5 t, y = 3 + t, z = 3t in symmetric form.. Write the equations of the line through (7,, ) and (3,, 4). 3. Write the equations of the line parallel to x (4,, 3). = y+4 = z 3 and through 4. Write the equations of the line through (,, 8) and perpendicular to the plane 3x + 7y z + = 0. 5. Write the equations of the line through (3, 4, 0) and perpendicular to the xy-plane. 6. Find the equation of the plane passing through the three points (,, ), (,, ), (, 0, ). 7. Find the equation of the plane perpendicular to the line joining (,, 0) and (3,, 5) and passing through (, 5, 8). 8. Reduce 3x + y z + 5 = 0 to normal form. 9. Reduce x 5y + z 3 = 0 to intercept form and write the coordinates of the intercepts. 0. Write equation of the plane passing through (,, ) and (3, 5, 4) which is perpendicular to the xy-plane.. Find the equation of the plane perpendicular to the plane x y + z 5 = 0, parallel to the line whose direction cosines are,, 5 and passing 5 5 5 through (, 4, ).. Find the equation of the plane parallel to the 3x y + 6z + 5 = 0 and passing through (, 4, ). 3. Find the equation of the plane parallel to the xz-plane through (, 4, 6). 4. Find the equation of the plane parallel to the xy-plane and units from it.
5. Find the equation of the plane with intercepts,, 4. 6. Find the distance from the plane x + y z + = 0 to the point (0,, 3). 7. Determine the angle between x + y z + 7 = 0 and 3x y 6 = 0. Bedia Akyar Moller
Analytic Geometry MAT 035 7.0.04 WEEKLY PROGRAM 7-8 We have started the chapter plane and given the vector form of the equation of a plane, the angle between two planes and we have also given other forms of the equation of a plane. I have given the distance from a point to a plane. We have mentioned some situations between planes and lines. We have talked about the intersections of two planes and some examples by different methods. We HAVE NOT SEEN intersections of three planes and intersections of lines and planes and the specialized distance formulae and finish the chapter plane. YOU ARE RESPONSIBLE OF THE CHAPTER PLANE INCLUDING INTERSECTION OF TWO PLANES FOR THE MIDTERM EXAM. GOOD LUCK WITH THE EXAM. After the midterm exam, we will mention some situations between planes and lines we will see intersections of three planes and intersections of lines and planes and finally the specialized distance formulae and finish the chapter plane. If time permits, we will start the new chapter Locus Problems. We will also give circles determined various conditions. We will do some examples. I highly recommend you to read Matematik Dunyasi, 005 Yaz, Guz, Konikler. Bedia Akyar Moller
Analytic Geometry MAT 035 4..04 WEEKLY PROGRAM 9-0 We talked about the chapter Plane. We have seen the equations of the planes which are perpendicular to the xy, xz and yz planes and also perpendicular to the coordinate axes. We have talked about the angle between two planes. I have also given the distance from a point to a plane (Please read the distance from a point to a line). We have seen the normal form of the equation (please read the normal form of a straight line). Although I have mentioned the situations between planes and lines I may again talk about the intersection of two planes and some examples by different methods (if it is necessary). We will see intersections of three planes and intersections of lines and planes and finally the specialized distance formulae and we will be finished the chapter plane. We will give the general form of a circle and also circles determined various conditions with aid of some examples. We are aiming to give intersections involving circles. I highly recommend you to read Matematik Dunyasi, 005 Yaz, Guz, Konikler. Do the following exercises: ) Find the vector formulation for the line of intersection of the planes P : x + y + 3z + 4 = 0 and P : x y + z = 0. ) Determine l P if l and P are given by l : (x, y, z) = (t +, t, 3t + ) and P : x y + z + = 0. 3) Find the distance from P 0 (3, 3, ) to the plane P : x + y z + 8 = 0. 4) Find an equation of the plane through (,,3) and perpendicular to the line x l = {(x, y, z) : = y+ = z }. 3 5) Find an equation of the plane through (,, 3), (,, ) and (-,,-3). x 6) Find an equation of the plane through (,, 3) and l = {(x, y, z) : = y+ = z }. 3 7) Find an equation of the plane determined by the lines l = {(x, y, z) : x + = y + = z} and l = {(t, t, 3t + 3) : t R}. 8) Determine the angle between the planes x+y z+7 = 0 and 3x y 6 = 0. 9) Find the cosine of the angle between the two planes with equations a) P : x + y + z = 0 and P : x y + z + 4 = 0, b) P 3 : x + y + z = 0 and P 4 : 4x y z 3 = 0, c) P 5 : x 6y + z = 0 and P 6 : x y + z 3 = 0.
0) Find a vector equation of the line of intersection of the planes in each case in the previous exercise. ) Find the equation of the plane parallel to the plane P : x 3y+7z = 0 and passing through (-, 3, 4). ) Find the equation of the plane perpendicular to the line joining (, 3, 5) and (4, 3, ) at the midpoint of these two points. 3) Determine α such that the following four planes P : x + y 3 = 0, P : 3x + 5z + 5 = 0, P 3 : x + y z 9 = 0 and P 4 : 3x + αy 5 = 0 which pass through the same point. Find the coordinates of this point. 4) Find a vector equation of the line through (,, 3) and in the direction of the vector u = (,, 3). 5) Find a vector equation of the line through the points (,, 3) and (,-, 3). 6) Find parametric equations of the line through (,,3) and parallel to the line through (,-,3) and (,-,4). 7) Find a symmetric equation of the line through (,,) and perpendicular to the two lines l = {(x, y, z) (t, t +, t)} and l = {(x, y, z) ( t, t, 3t + 3)}. 8) Find l l for given in the previous exercise. 9) Find the distance from P 0 (, 0, ) to the line l = {(x, y, z) (t, t +, t + )}. 0) Determine the locations (i.e. whether they are on the same side with the origin or not) of the points A(4,5,4), B(-5,8,-6) and C(3,5,4) with respect to the plane P : x y + 3z 7 = 0. ) Find the distance from (,, 3) to a) the line l = {(x, y, z) : x = y+ = z 3}. b) the plane P : x + y + z = 0. ) Find the distance from (, 0, ) to the line = {(x, y, z) : (x, y, z) = (t, t +, t + ), t R}. 3) Determine α such that the following four planes P : x + y 3 = 0, P : x + y z 9 = 0, P 3 : 3y + 5z + 5 = 0 and P 4 : 3x + αz 5 = 0 which pass through the same point. Find the coordinates of this point. 4) Determine l P if l and P are given by l = {(t +, t, 3t + ) : t R} and P = {(x, y, z) : x y + z + = 0}. 5) Determine P if and P have equations: a) l = {( t, t+, t ) : t R} and P = {(x, y, z) : x y +3z +4 = 0}. b) l = {(x, y, z) : x = y+ 0}. = z 3 } and P = {(x, y, z) : x + y + 3z 4 =
c) l = {(x, y, z) : x+6 3 = y = z } and P = {(x, y, z) : x y+z 4 = 0}. x 6) Given l = {(x, y, z) : = z }, find a line 3 l through (,, 3) such that l and l are skew. 7) Discuss the intersection of given planes with respect to the parameter λ. P : (λ + 5)x + (λ + 0)y + 3z 7 = 0, P : x + (λ + 7)y + z + 5 = 0. 8) Discuss the intersection of given planes with respect to the parameters a and b. P : x + y z + b = 0, P : x y + 3z = 0, P 3 : x + ay 6z + 0 = 0. 9) Find the distance between the lines l = {(x, y, z) : x + y z + 3 = 0, x y 4z+ = 0} and l = {(x, y, z) : 3x+y+5z = 0, y+z+ = 0}. 30) Find the equation of the circle tangent to the two axes and passing through the point (, 7). 3) Find the equation of the circle which is tangent to the y-axis, and passes through the point (, ) and the center of which is on the line x + y + 4 = 0. 3) Find the equation of the circle which passes through the points (, 4) and (4, 6) and whose center lies on the line 3x y + 0 = 0. 33) Show that for any value of θ the point (rcosθ, rsinθ) lies on the circle x + y = r. 34) Find an equation of the line which is tangent to the circle x + y + 4x 6y + 8 = 0 at the point ( 3, 5). 35) Find the equation of tangent line to each circle at the point indicated: a) x + y = 3, (, ). = y+ b) (x ) + (y + ) = 9, (, ). c) x + y + 4x 5y + 9 = 0, (, 3). 36) Find the equations of the tangents to the circle x + y = 6 drawn from the point ( 3, 7). 37) Find the equations of the tangents to (x + ) + (y ) = 9 with slope. 38) Find any points of intersection of curves a) x + y + 4x 6 = 0, x + 3y 0 = 0. b) x + y + 8x 6y = 0, x + y + = 0. 39) Find an equation of each tangent line drawn from the point (5, ) to the circle x + y = 3 by considering all possible methods. 40) Discuss the intersection of given lines and circles below, and if they intersect each other then find their intersection point.
a) l : x y 3 = 0, C : x + y 3x + y + 3 = 0. b) l : 3x y + 5 = 0, C : x + y + 4x y + 5 = 0. c) l : 3x y + = 0, C : x + y + x + y + 8 = 0. 4) Find any points of intersection of the given curves and an equation of radical axis, and draw a picture. a) x + y + x = 0, x + y y = 0. b) x + y + 8x 6y = 0, x + y + 3x y 4. c) x + y + 4x y = 0, x + y + 3x y 4 = 0. 4) Show that each of circles x + y = 9, x + y x + 7 = 0 and x + y 6x 8y + = 0 is tangent to other two. Do the common tangents meet in a point? If they do, find the point. 43) For which value of the parameter µ, the length of the tangent drawn from the point A(5, 4) to the circle C : x + y + µy = 0 is? 44) Show that the circles C : x + y 6x y + = 0 and C : x + y 4x + 4y + 6 = 0 intersect each other under right angle. 45) Discuss the intersection followings with respect to the parameter λ. a) l : λx y + = 0, C : x + y 0x + 4 = 0. b) C : x + y 3x + 8y 5 = 0, C : x + y + 6x y + λ = 0. Bedia Akyar Moller
Analytic Geometry MAT 035 08..04 WEEKLY PROGRAM - I highly recommend you to read Matematik Dunyasi, 005 Yaz, Guz, Konikler. I am going to give another locus problem called Ellipse. I will give parametric equations of an Ellipse and the standard form of the equation of an ellipse. I am going to give last two locus problems called Hyperbola in the standard form and parabola, chord and the latus rectum of a Parabola. If time permits, after finishing the 4th chapter, I will start the next chapter and give Transformation of Coordinates. I am going to give translation of axes, rotation of axes and do some examples. Do the following exercises:. Find an equation of the indicated ellipse a) Foci ( 3, 0), a = 5. Also find the focal radii of a point for which x =. b) Foci (0, ), a = 4. c) Vertices ( 4, 0) and passing through the point (3, 5).. Find the coordinates of the foci and the vertices, and draw the graph of the equation a) 9x + 4y = 36. b) x + 4y = 6. c) x + 6y = 9. 3. Find an equation of the specified curves a) Of the locus of points the sum of whose distances from the points (0, ) is 6. b) Of the locus of points the sum of whose distances from the points (, 3) and (4, ) is 8. 4. Find an equation of the indicated hyperbolas.
a) Foci ( 5, 0), a = 3. Also find the focal radii of a point for which x = 6. b) Foci ( 3, 0), vertices (, 0). Also find the focal radii of a point for which x = 4. c) Foci (0, 4), b =. d) Foci (0, 5), ends of conjugate axis ( 3, 0). e) Vertices (0, 3) and passing through the point ( 3, 3). f) Foci ( 5, 0)and passing through the point ( 0 3, 4). g) Vertices (, 0) and asymptotes with slopes 3. h) Foci (0, 4) and (0, 0), and passing through the point (3, 4). 5. Find the foci, vertices, and asymptotes, and draw the graph of the equations: a) 6x y = 6. b) 8x y = 8. c) 4x 9y + 36 = 0. 6. Find an equation of the specified curves: a) Of the locus of points the difference of whose distances from the points (0, ) is. b) Of the locus of points the difference of whose distances from the points ( 4, 0) is 6. c) Of the locus of points the difference of whose distances from the points (, 3) and (4, ) is. 7. Find equations of asymptotes of the given hyperbolas in the form of single second degree equations: a) x 3y + 4 = 0. b) x y = 0. c) x 4 y 9 = 0. 8. Find an equation of the indicated parabolas. a) Directrix x =, focus (, 0). b) Directrix y = 4, focus (0, 4). c) Vertex (0, 0), directrix y = 4 3.
d) Vertex (0, 0), directrix x = 3 4. e) Vertex (0, 0), focus on the x axis, and passing through the point (8, 4). 9. Find the coordinates of the focus and equation of the directrix for the given parabolas and sketch the graphs. a) y = 6x. b) 3x + 4x = 0. c) x 8y = 0. d) 3x + y = 0. 0. Find the specified equations. a) Of the locus of points equidistant from the line y = 4 and the point 3 (0, 4). 3 b) Of the locus of points equidistant from the line x = 5 and the point ( 5, 0). c) Of the parabola with y = as directrix and (0, 4) as focus. d) Of the parabola with y = 3 as directrix and (, 0) as focus. e) Of the parabola with x + y + = 0 as directrix and (, 3) as focus. f) Of the parabola with 3x 4y = as directrix and (3, 0) as focus.
Analytic Geometry MAT 035..04 WEEKLY PROGRAM 3 After finishing the 4th chapter, I have started the next chapter and given Transformation of Coordinates. We have already talked about translation of axes, rotation of axes and done some examples. Now I am going to start the new chapter Equations of Second degree. I am going to give classification of conics and conic sections. We are going to discuss all possible cases of conic sections. If time permits I will also talk about the family of conics. Do the following exercises:. a) Translate axes to the new origin (, 3) and reduce the equation of the curve x 4y 4x + 4y 36 = 0. b) For the equation x + 4xy + y = rotate the axes through θ = 45.. The x, y axes are translated to x, y axes with origin O (, 3). Find x, y coordinates of the point (4, 6). 3. Transform the equation 4x y 6x 7y 8 = 0 under the translation defined in Exercise. 4. Find x, y coordinates of the point (9, 5) under the rotation of axes through the obtuse angle θ for which tanθ = 3 4. 5. Transform the equation 3x 5y = under the rotation defined in Exercise 4. Bedia Akyar Moller