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Vector Calculus

Vector Spaces and Projection Mappings

Please see the attached file for the fully formatted problems. Let V be a vector space of all real continuous function on closed interval [ -1, 1]. Let Wo be a set of all odd functions in V and let We be a set of all even functions in V. (i) Show that Wo and We are subspaces and then show that V=Wo⊕We. (ii) Find a pro

Vector Cross Product and Arc length

1 Given a = <4, -3, -1> and b = <1, 4, 6>, find a X b. 2 Find the arc length of the curve given by x = cos 3t, y = sin 3t, z = 4t, from t = 0 to t = pi/2.

Vectors, Work and Orthogonal Vectors

Find the small positive angle from the positive X-axis to the vector OP that corresponds to (3,3) Find the work done by the constant force 4i-2j if the point of application moves along the line segment from P(1,1) to Q(5,7) Determine m such that the two vectors 3i-9j and mi+2j are orthogonal

Vector Fields : Divergence, Scalar Curl and Gradient

1) Given a vector field P(x, y) i + Q(x, y) j in R2, its scalar curl is the k - component of the vector curl (Pi + Qj + 0k) in R3 (the i and j components of that curl are 0). In other words, the scalar curl of the vector field Pi + Qj equals &#8706;Q/&#8706;x - &#8706;P/&#8706;y. Find the scalar curl of F(x, y) = sin (xy) i +

Finding the Vertex of a Parabola : Maximum Height

An arrow is accidentally shot into the air. The formula y = - 14x2 + 56x + 18 models the arrow's height above the ground, y, in feet, x seconds after it was released. When does the arrow reach its maximum height? What is that height? keywords: quadratic equations, vertex form

Vector Spaces and Dimensions

Problem 1. Let V be a finite-dimensional complex vector space. Then V is also a vector space over real numbers R. Show that dimV ( over R) = 2*dimV(over complex C). Hint: If B={v1, v2, ..., vn} is a basis of V over C, show that B'={v1, ..., vn, iv1, ... ivn} is a basis of V over R. Problem 2. ( extend problem1) Let L be a f

Vector Spaces and Scalar Multiplication

1)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. Prove axioms (VS1)=For all x,y, x+y=y+x (commutativity of addition), (VS3)= There exist an element in V denoted by 0 such that x+0=x for each x in V.,(VS4)= For each element x in V there exist an element

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

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.


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.


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 spac

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