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    Vector Calculus

    Vecor Spaces and Linear Combinations

    Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication. 1. Label the following statements as true or false

    Vertex of a Parabola

    A football is thrown, by a quarterback, to a receiver 40 yards away. The formula y = -0.025x^2 + x + 5 models the football's height above the ground, y, in feet, when it is x yards from the quarterback. How many yards from the quarterback does the football reach its greatest height? What is that height? keywords : find,

    Vector spaces and proof of theorem

    Let V be a vector space over an infinite field F. Prove that V cannot be written as the set-theoretic union of a finite number of proper subspaces. I know that I should use proof by contradiction to prove this statement, but need help from there.

    Vectors : Addition, Speed and Direction

    Find |a|, a + b, a ? b, 2a, and 3a + 4b. 18. a=21?3j. bi+5j 19. a=3?2k, b=i?j+k 31. A woman walks due west on the deck of a ship at 3 mi/h. The ship is moving north at a speed of 22 mi/h. Find the speed and direction of the woman relative to the surface of the water.

    Vectors : Work Done

    Determine the work done by a force of F Newtons acting at a point A on a body, when A is displaced to a point B, the coordinates of A and B being (2,5,-3) and (2,-3,0) metres respectively, and when F=2i-5j+4k Newtons. Please use a diagram to illustrate your answer.

    Vectors : Force and Moment

    A force of magnitude 3 units act at the point with coordinates (1,2,3). The force is applied in the direction of the vector 3i-1j+4k. find the moment of the force about O (the origin). What is the moment of the force about the point with coordinates (1,2,3). please use diagram to illustrate your answer.

    Example of a quadratic model

    QUADRATIC MODELING: You will need to locate data that can be modeled using a quadratic function. Keep in mind that good candidates for quadratic models have data that both increases and decreases. Once again, I encourage you to use either online or print resources, and I would also refer you to the textbook website which has

    Vector Spaces, Basis and Closest Vector

    1. Let S be a subset of R described as follows: S {(x,y,z) :x+y+z = 0} (a) Show that S is a vector space. (b) Calculate a basis of S and compute it dimension. (c) Find the vector in S which is closest to the vector (1,3, ?5) in R3.

    Inner Space Proof

    Question with topic of orthogonal in inner product. See attached file for full problem description.

    Vectors, Basis, Row Space, Column Space and Null Space

    1. Which of the following sets of vectors are bases and why are they bases for P2 A) 1-3x+2x^2, 1+x+4x^2, 1-7x B) 4+6x+x^2, -1+4x+2x^2, 5+2x-x^2 C) 1+x+x^2, x+x^2, x^2 2. In each part use the information in the table to find the dimension of the row-space, column-space and null-space of A and the null space

    Parallel and Perpendicular Vectors and Work

    1 Given a = 9i - 5j and b = 7i-4j, express i and j in terms of a and b 2 Given a=<4,5,-3> and b =<4,-2,2> determine whether a and b are parrallel, perpendicular, or neither. 3 Given F = 4i -2k;..... P(0,1,0) and Q(4,0,1) find the work W done by the force (F)moving a particle in a straight line from P to Q. 4 Given a

    Laplace Operators and Gradient Vectors

    Let f(z)=u+iv be an analytic function, phi(u,v) any function with second order partial derivatives and g(u,v) any function with first order partial derivatives. a) Let L_x,y be the Laplace operator in x,y coordinates and L_u,v be the Laplace operator in u,v coordinates. Show that L_x,y(phi o f)=L_u,v |f'(z)|^2 b)Let G_u,v be t

    Nonzero Vector Projections

    Let U and a be nonzero vectors Show explicitly that the the angle between: Proj a U = Projection of vector U along vector a and The vector U - Proj a U (vector component of U orthognal to a) is a right angle (i.e show that the dot product of these two vectors is 0)

    Vector Spaces, Mappings and Kernels

    Let V1 and V2 be finite dimensional vector space over K and let SєL(v1, V2). Prove that there exists T є L(V2, V1) such that S =STS and T= TST. See the attached file.

    Equation of a Plane

    Find an equation of the plane through the point (-2, -4, 0) and perpendicular to the vector (-5, -5, 0).

    Writing Equations from Word Problems

    Two planes left an airport at noon. one flew east at a certain speed and the other flew west at twice the speed. The planes were 2700 mi. apart in 3 hours. how fast was each plane flying? Write an equation and solve.

    Vectors and Grad Proof

    If V is a vector function, show the following by expansion that the following equality may be maintained: (V.grad)V=(gradxV)xV + grad(V^2/2) Please see the attached file for the fully formatted problems.

    Express each vector as linear combination of basis.

    Find a subset of the vectors that forms a basis for the span of the vectors; then express each vector which is not in basis as a linear combination of the basis vectors. v1= (1, 1, -1), v2= ( 1, 0, 1), v3= ( 1, -2, 5), v4= ( 5, 3, 2)

    Period of a vector function

    I need assistance finding the period of a vector valued function. The function is of the form cos^2(2T)+Sin^2(3t) + cos(2t-pi/2) I remember doing something like this back in pre calc where the period was 2pi/b but the functions were not vectors. How would I proceed from here? Any help is appreciated.