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    Vector Calculus

    Vector Proofs : Cross Products and Dot Products

    15. (a) Use a counterexample to show that (A x B) x C is not necesarilly equal to A x (B x C) toAx(x). (b) Prove that A x (B x C) = (A.C)B ? (A. B)C. Can von give a geometric interpretation of this equation?

    Prove that U is a Subspace of V and is Contained in W

    Please view the attached file for the full solution. What is presented below has many missing parts as the full question could not be copied properly. Let F be the field of real numbers and let V be the set of all sequences: ( a_1, a_2, ... , a_n, ... ), a_i belongs to F, where equality, addition and scalar multiplicat

    Let F be the field of real numbers and let V be the set of all sequences ( a_1, a_2, ... , a_n, ... ), a_i belongs to F, where equality, addition and scalar multiplication are defined component wise. Prove that V is a vector space over F.

    Let F be the field of real numbers and let V be the set of all sequences ( a_1, a_2, ... , a_n, ... ), a_i belongs to F, where equality, addition and scalar multiplication are defined component wise. Prove that V is a vector space over F. See attached file for full problem description.

    Equation of plane

    Find the equation of a plane through the origin and perpendicular to: x-y+z=5 and 2x+y-2z=7

    Vectors in Component From

    Given points A(3, -4) and B(-5, -2): 1. Express AB in component form (-2, -6) 2. Express BA in component form -2, 3. Find |AB| and |BA|

    Euclidean Spaces and Subspaces

    1. Let W = {(a, b ,2a - 3b, -a + 2b)} whre a and b are real numbers. (a) In what Euclidean space does this subset reside? Explain your answer. (b) Show that W is a subspace by showing that it satisfies the closure properties. (c) Show that W is a subspace by describing W as the span of a set of vectors. (d) Explain why this

    Orthogonality, Orthonormal Basis and Orthogonal Complement

    Please help me with 2 attached problems. In problem 6 I can imagine the space and guess which vectors are orthogonal to (1,-1,0) like (0,0,1) for example. However, I am not sure of my approach when I do it by hand. I do not know how to approach problem 12. I am doing something completely wrong and especially need help with

    Vectors and Curvature

    Find k(t) for y = 1/x. ^ P.S. The curvature of a curve is k = dT/ ds, where T is the unit tangent vector. ^ ^ And k(t) = T'(t)/ r'(t)

    Vector Space Theorems and Matrices

    2. Use Theorem 5.2.1 to determine which of the following are subspaces of M22. Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold. (a) If u and v are vectors in W, then u + v is in W. (b) If k is any scalar and u is any vector in W,

    Vector Functions : Velocity, Speed and Acceleration of a Particle

    Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and thaw the velocity and acceleration vectors for the specified value of t. 3, r(t)= <t^2 ? 1, t>, t=1 4. r(t)=<2?t, 4 sqrt t), t=1 5. r(t)=e^t i+e^-t j, t=0

    Particles, Forces and Vectors

    See the attached file. 5. A particle P is acted on by three forces F1, F2 and F3, where F1 = (2i - 5j)N and F2 = (4i - 4j). Given that P is in equilibrium, a. Find F3 in terms of i and j The force F3 is now removed and P moves under the action of F1 and F2 alone. b. Find to 3 s.f. the magnitude of the resultant force

    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&#8853;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.

    Properties of the determinant function

    Please see the attached file for the fully formatted problems. 2. Verify that det(AB) = det(A) det(B) for A = 2 1 0 and B = 1 -1 3 3 4 0 7 1 2 0 0 2 5 0 1 Is det

    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