[2 pointsLet A= a b 3 20 620 6 Find all possible values of a and b for which A is a symmetric matrix. [3 points] Fill in the blanks with MIGHT, CANNOT, or MUST. Give a brief explanation of your answer. (a) If the columns of A are linearly independent, then AX = B have a unique solution. (b) If A is a square matrix and AX =B has only the trivial solution, then det(A) equal 0. form a...
2. Let A:(-1,1,-1), B:(2,0.2), C:(4.1.-3), and D:(-3, 1, 10) be points in R. (a) Find the angles (in degrees) of the triangle with vertices A, B and C. (b) Find an equation of the plane passing through the points A, B, and C. (c) Find two unit vectors perpendicular the plane through A, B, and C. (d) Find the volume of the tetrahedron with vertices A, B, C, D. 3. (a) Find an equation of the tangent line to the...
3. Let S be the plane through the points A(0,0,-1), B(0,2,0), C(3,0,0). (a) Find an equation for S of the form ax +by+cz = d. (b) Find the point on that is closest to the point P(2,2,3).
1. (10) Let l be the line in 3-space that passes through the points A=(5,2, -1) and B = (6,0,–7). (a) Find a set of parametric equations for l. (b) Find the unique point P at which l intersects the plane with equation -3.21 + 722 - 2.23 = 11. (c) Let P be the point found in part (b), and let Q = (k, 7, 10) for an unspecified real number k. Determine the value of k for which...
vectors (b)) Let a, b and c be three vectors such that a is perpendicular to both b and and Ibl = lel. Showr that the equation of the plane through the three points whose position vectors are g, b and c, is le尸. }blicl+b.ef Hence find the equation of the plane through the points (2,-1,1,).(3,2,-1).(-1,3.2) ㄈ vectors (b)) Let a, b and c be three vectors such that a is perpendicular to both b and and Ibl = lel....
1. Let ū= (2,4,-1), v = (3.-3,-1) (a) Compute: x ū (b) Compute: ü x 7 (c) Is the cross product commutative? If not, what is it instead? 2. Let A = (7, -11,3), B = (1,9, -3), C = (-6,3, -2), D= (0,-8, 12), E = (1, -13,2) (a) Give the vector equation of a line passing through the points A, B. (b) Find the equation of the plane containing the points C,D,E. (c) Find the point of intersection...
Question 4 6-69) Consider the following three points: a-0 b-3and c-1 a) Find the parametric vector equation of the plane which passes through these three [3 marks] b) Find the vector cartesian equation of the plane passing through the three points listed [2 marks] c) Hence, or otherwise, find the non-vector cartesian equation of the plane passing through 3 marks] points above. the points above. Question 4 6-69) Consider the following three points: a-0 b-3and c-1 a) Find the parametric...
(3) 9. (a) Let (21,91,2), B(22,92,-2),(3,3,3) be three non-collinear points in that is, three points which do not all lie on a straight line. Then the equations of the plane through these three points is: 30 Iii 0 12 32 22 3 Page 1 2 (b) Find the equation of the plane through (1,2,2), B(1,2-1), B(0,1,2)
Consider the null cone of the three-dimensional Minkowski space (R2+1,m A. Write the equation of N in standard coordinates (t,,2) of R2+1 B. Let p (a,b,c) be a point (not the origin) on M. Draw the tangential plane to N at p. Moreover, draw all null vectors with origin at p. Consider the null cone of the three-dimensional Minkowski space (R2+1,m A. Write the equation of N in standard coordinates (t,,2) of R2+1 B. Let p (a,b,c) be a point...
Question 3 a) Find the cartesian equation of the line that passes through the origin and lies perpen- (3 dicular to the plane 3x - 5y +2z 8. marks] b) Find the cartesian equation of the plane that lies perpendicular to the line 3 marks] and passes through the point 1 cExplain why a unique plane passing through the three points A(-2,-1,-4), B(0,-3,0) [2 marks) and C(2,-5,4) cannot be defined. Question 3 a) Find the cartesian equation of the line...