Revision Problems for Examination 2 in Algebra 1

Transcription

1 Centre for Mathematical Sciences Mathematics, Faculty of Science Revision Problems for Examination in Algebra. Let l be the line that passes through the point (5, 4, 4) and is at right angles to the plane x + y + z = 7. Determine the shortest distance from the point P = (,, ) to the line l and state specifically the point on l closest to P. (The coordinate system is assumed to be orthonormal.). Find a positively oriented orthonormal basis e, e, e such that the vector e is parallel to the line (x, y, z) = ( + t, 7, t) and the vectors e and e are both parallel to the plane x y + z =. In how many ways can such a basis be chosen?. Solve the system of linear equations x + y + (a + )z = a + (a + )x + ay + z = a + ax + (a )y + (a )z = a for all values of the real number a for which this is possible. 4. Let M be the plane through the points (, 4, 0), (,, 0) and (0,, ). The sphere with centre at the point (,, ) and radius 5 intersects the plane M along a circle. a) Determine an equation in normal form for the plane M. b) Calculate the shortest distance from the centre of the sphere to M. c) Determine the centre and the radius of the circle of intersection. (Positive orthonormal system.) 5. Let P = (,, 0) and Q = (0,, 0) and let R be a point in the xz-plane at the distance to each of the points P and Q. Let l be the line through the point P at right angles to the plane through P, Q and R. Determine all points S l such that the volume of the tetrahedron P QRS is 4. (Positive orthonormal system.) 6. Consider the matrix 0 A = a) Compute the inverse A of A. b) Show that A A = A E. (E denotes the unity matrix.) c) Find an expression for A n A n for all natural numbers n.

2 7. Solve the system of linear equations x + y + z = 5 x y + az = x + y + z = b for all values of the real numbers a and b. 8. Let l be the line of intersection of the planes x y = 7 and 6x y z = 4. a) State a parametric equation for l. b) Show that l is parallel to the plane M : y z =. c) Compute the distance from l to the plane M. (Orthonormal system.) 9. Let l denote the line obtained when the line l 0 : (x, y, z) = ( t, t, 5 t) is projected orthogonally onto the plane M : x + y + z = 5. State a parametric equation for l. (Orthonormal system.) 0. A triangle has two of its corners at the points (, 0, ) and (0,, ). Find the minimum value of the area of the triangle when the third corner runs through the line with equation (x, y, z) = (,, t), t R. (Positive orthonormal system.). Let P = (5,, ), Q = (,, 7) and R = (, 6, ) be three points in space. (Positive orthonormal system.) a) State an equation in normal form for the plane through the points P, Q and R. b) Determine the point of intersection of the perpendicular bisectors of the sides P Q and P R of the triangle P QR. c) State a parametric equation for the line l in space the points of which are at the same distance to the points P, Q and R.. Consider the matrix 0 A = a) Compute the inverse A of A. b) Compute A, A and A 4. c) Find a formula for A k for k N, k, and verify it by induction.. Show that the vectors u = (, 0, ), u = (0,, ) and u = (,, ) form a basis for -space and determine the coordinates of the vector v = (5, 6, ) with respect to this basis. 4. State a parametric equation for the plane M through the points (,, ), (,, 0) and (, 6, 4) and compute the minimum distance from the point (0,, 0) to M. (Positive orthonormal system.) 5. Let L be the line of intersection of the planes x + y + z = and x y + z = 6

3 and let M be the plane through the origin that is at right angles to the line (x, y, z) = (,, )t, t R. State a parametric equation for the line L and show that L intersects the plane M in a point. Also find the point of intersection. (Orthonormal system.) 6. Determine for each real number a the number of solutions of the system ( a)x + ( a)y + z = + a x + y + ( a)z =. ( a)x + ( a)y + z = Solve the system completely for those values of a for which this is possible. 7. Let P = (,, ) and Q = (,, ) be two points in space (positive orthonormal system). a) State a parametric equation for a line l that is at right angles to the line segment P Q and passes through the midpoint of that segment. b) Determine a point R on the line l through the points (,, ) and (, 0, 9) such that the triangle P QR is isosceles with the sides RP and RQ of equal length. c) Compute the area of the triangle P QR. 8. Let A, B and C be the following matrices: A =, B =, C = 6 9. a) Determine whether the matrices A, B and A B are invertible and state the corresponding inverse when applicable. b) Show that AB = BA and that (A + B) n = A n + B n for each natural number n. c) Compute (A + B) n for each n =,,.... d) Solve the matrix equation AX BX = C.

4 Answers. The distance is, the point is (,, ).. E.g. e = (, 0, ), e = (,, ), e = 6 (,, ). There are four such bases.. There is a unique solution (x, y, z) = (a, a, ) when a and there are infinitely many solutions (x, y, z) = (t, t +, t), t R when a =. 4. a) x + y z = 6. b). c) Centre at (,, ), radius R = (, 0, ) gives S = (,, ) or S = (,, ). R = (, 0, ) gives S = (,, ) or S = (,, ). 6. a) A = c) A n A n = A E for all natural numbers n. 7. The system has a unique solution (x, y, z) = ( b 5, 0a + b ab 8, a + ) 8 b a + when a. When a = and b = 9 the system has infinitely many solutions (x, y, z) = (4, t, t), t R. When a = and b 9 there are no solutions. 8. a) (x, y, z) = (t, t 7, 4t 7), t R. c) 8/ E.g. (x, y, z) = (,, 4) + t(,, ), t R. 0. /.. a) x + y + z = 0. b) ( 7, 0, ). c) (x, y, z) = ( 7, 0, ) + t(,, ), t R.. a) b) A = 0 A = , A = 0 0, A 4 =

5 c) A k = k k(k ) 0 k 0 0, k =,,,..... u = (,, ) with respect to the basis u, u, u. 4. E.g. (x, y, z) = ( + s, + s + 4t, s + t), s, t R (x, y, z) = ( 9 + t, t, 8t), t R. (0,, ). 6. The system has a unique solution when a 0 and a, has no solution when a = and has infinitely many solutions when a = a) E.g. (x, y, z) = (, + t, ), t R. b) R = (5,, 5). c) a) A and B are not invertible, A B is invertible with inverse E. b) AB = BA = 0. c) (A + B) n = n E when n is even, (A + B) n = n (A + B) when n is odd. d) X =. 5