Homework Answers
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14) Consider the parallelepiped D determined by the vectors (2,-1,2), (1,3, 1), and (2,-1,1). Let...
14) Consider the parallelepiped D determined by the vectors (2,-1,2), (1,3, 1), and (2,-1,1). Let T(z, y, 2)a-ytz. Consider the integral I - JSsD TdV. Using the Change of Variables Theorem, write I as an integral of the form T(r(r, s,t), v(r, s, t), z(r, s,t))lJ(r,s, t) dr ds dt for a suitable linear change of variables (r, s, t) (, y,z). The Jacobian J(r,s,t) you get here should be a constant function.
14) Consider the parallelepiped D determined by...
(1) Let 7 =< 2,1,-2 > and 7 =< 1,2,3 >. Find two vectors and such...
(1) Let 7 =< 2,1,-2 > and 7 =< 1,2,3 >. Find two vectors and such that ✓ = 7+7, where is parallel to 7 and is orthogonal to 7.
b. Find the volume of the parallelepiped spanned by the vectors (t, 0,0), (1,2,-4), (0, t,-1). For what values of t will there be a zero volume? What can you say about the three vectors when the volu...
b. Find the volume of the parallelepiped spanned by the vectors (t, 0,0), (1,2,-4), (0, t,-1). For what values of t will there be a zero volume? What can you say about the three vectors when the volume is zero? Using a 3D graphing program, include two graphs of the three vector, one where the volume is not zero and one where the volume is zero. (9pts)
b. Find the volume of the parallelepiped spanned by the vectors (t, 0,0),...
To 17. Determine whether the vectors f(1,2,3), (1,-1,2), (1,-4,2)) in R3 are linearly independent.
To 17. Determine whether the vectors f(1,2,3), (1,-1,2), (1,-4,2)) in R3 are linearly independent.
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle betwe...
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
Let the universal set S be S = {1,2,...,10}, and A = {1,2,3}, B = {3,4,5,6,7}...
Let the universal set S be S = {1,2,...,10}, and A = {1,2,3}, B = {3,4,5,6,7} and C = {7,8,9,10} 1) Find (A∪C)−B 2) Find A^c ∩(B^c ∪C)
vectors (b)) Let a, b and c be three vectors such that a is perpendicular to both b and and Ibl = lel. Showr that th...
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....
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
roblem 1: Consider the set of all vectors in R1 which are mutually orthogonal to the vectors <3,4,-1,1> and (a) The first thing you need to do is determine the form of all vectors in this space...
roblem 1: Consider the set of all vectors in R1 which are mutually orthogonal to the vectors <3,4,-1,1> and (a) The first thing you need to do is determine the form of all vectors in this space. Hints on how to proceed You need vectors < a,b,c,d> with the property that <a,b,c,d> is orthogonal to <3,4,-1,1>and <a,b,c,d is orthogonal to <1,1,0,2>. There's a vector equation that defines "orthogonal" and this will set up two equations. .That means you have two...
Determine whether the given pair of vectors are orthogonal (perpendicular) (a) a2,0,1> and b=< -4,3,8> (b)...
Determine whether the given pair of vectors are orthogonal (perpendicular) (a) a2,0,1> and b=< -4,3,8> (b) a<1,-4, 6 > and 6 =< 1,-1,1 >
14) Consider the parallelepiped D determined by the vectors (2,-1,2), (1,3, 1), and (2,-1,1). Let T(z, y, 2)a-ytz. Consider the integral I - JSsD TdV. Using the Change of Variables Theorem, write I as an integral of the form T(r(r, s,t), v(r, s, t), z(r, s,t))lJ(r,s, t) dr ds dt for a suitable linear change of variables (r, s, t) (, y,z). The Jacobian J(r,s,t) you get here should be a constant function.
14) Consider the parallelepiped D determined by...
(1) Let 7 =< 2,1,-2 > and 7 =< 1,2,3 >. Find two vectors and such that ✓ = 7+7, where is parallel to 7 and is orthogonal to 7.
b. Find the volume of the parallelepiped spanned by the vectors (t, 0,0), (1,2,-4), (0, t,-1). For what values of t will there be a zero volume? What can you say about the three vectors when the volume is zero? Using a 3D graphing program, include two graphs of the three vector, one where the volume is not zero and one where the volume is zero. (9pts)
b. Find the volume of the parallelepiped spanned by the vectors (t, 0,0),...
To 17. Determine whether the vectors f(1,2,3), (1,-1,2), (1,-4,2)) in R3 are linearly independent.
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
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....
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
roblem 1: Consider the set of all vectors in R1 which are mutually orthogonal to the vectors <3,4,-1,1> and (a) The first thing you need to do is determine the form of all vectors in this space. Hints on how to proceed You need vectors < a,b,c,d> with the property that <a,b,c,d> is orthogonal to <3,4,-1,1>and <a,b,c,d is orthogonal to <1,1,0,2>. There's a vector equation that defines "orthogonal" and this will set up two equations. .That means you have two...
Determine whether the given pair of vectors are orthogonal (perpendicular) (a) a2,0,1> and b=< -4,3,8> (b) a<1,-4, 6 > and 6 =< 1,-1,1 >
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