### Vectors in the Plane and in Space : Distance Between Lines, Area of a Parallelogram and Intersection of Lines and Planes (8 Problems)

Please see the attached file for the fully formatted problems.

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Please see the attached file for the fully formatted problems.

#2) Flux integrals; s F?n dA. Evaluate the integral for the following data. Indicate the kind of surface. (show the details of your work): F=[x2, ey,1], S: x+y+z=1, x 0, y 0, z 0 Kreyszig's Advanced engineering mathmatics 8th ed.: section 9.6: Surface integrals Please s

Let be a vector space and a subset of such that implies and for Define a partial order on by defining to mean . A linear functional on is said to be positive (with respect to ) if for . Let be any subspace of with the property that for each there is an with . Assume that , where Then each positi

Can you please let me know how to approach those proof questions. Consider the polyhedron P = {x  Rn : xi > 0 for all i = 1 ... n}. a)Prove that the origin (i.e. the vector of all 0's) is a vertex of P, according to the definition of a vertex (i.e. do not rely on the fact that vertex = extreme point = basic feasibl

Let m be an area measure on {z in C:|z| < 1}. Show that 1, z, z^2,... are orthogonal vectors in L^2(m). Find ||z^n||, n >= 0. If e_n=(z^n)/||z^n||, n >= 0, is {e_0, e_1,...} a basis for L^2(m)?

Two opposite vertices of a square are P(3, 2) and Q(-5, -10). Find the length of: a) a diagonal of the sqaure b) a side of the sqaure

The vector v has an initial point p of (7,3) and terminal point q of (-5,-4). How do I find its position vector and write v in the form of ai + bj?

Given these vectors how do I find u - v? U=-8i-6j V=-10i+5j

Using the given vectors how do I find the specified dot product u=3i-8j;v=4i+9j find u.v

--- - In space, given 3 points M (1,0,0), P (0,2,0), Q (0,0,3). Let C be the contour... --- (See attached file for full problem description)

3. Two vectors Xand Y are said to be orthogonal (perpendicular) if the angle between them is r/2. (a) Prove that X and V are orthogonal if and only if X . Y = 0. Use part (a) to determine if the following vectors are orthogonal. (b) X =(?6,4.2) and Y =(6,5,8) (c) X=(?4,8,3) and Y=(2,5,16) (d) X = (?5. 7, 2) and Y = (4, 1, 6

Important Formulas and their Explanations (III): Gradient, Divergence and Curl Gradient of a constant

See attached

If v = i - j and w = i + j, find the dot product v * w and the cosine of the angle between v and w.

If v=3i-4j, find ||v||

4) A vector field F is shown. Use the interpretation of divergence derived in this section to determine whether div F is positive or negative at P1 and P2 a.) Are the points P1 and P2 so sources or sinks vector field F shown in the figure? Give an explanation based solely on the picture. 5)Use the divergence Theorem to c

B. A region is surrounded by two infinitely long concentric cylinders of radii, a1 and a2 (a2>a1). The concentric cylinders are charged to potentials phi1 and phi2 respectively. Determine the potential and electric field vector everywhere in the region. Please see the attached file for the fully formatted problem.

For each of the following vector fields F type in a potential function f (please see attached for full question)

Show ... is independent of path for any central force F....

Consider the force F shown below... [DIAGRAM] Select the option giving the i-component of F.

Hi, I cannot figure this problem out. I would like to see how to work it. The answer in the book is 3/e, but I cannot get it. There must be a trick involved (hence the author's hint to think carefully), but I'm not sure what it is. I attached the problem. Thanks

Please see the attached files for the fully formatted problems.

(a) Consider a vector function with the properly ... = 0 everywhere on two closed surfaces S1 und S2 and in the volume V enclosed by them (see the figure). Show that the flux ol F through S1, equals the flux of F through S2. In calculating the fluxes, choose the direction of the normals as indicated by the arrows in the figure.

(a) One of Maxwell's equations states that V H = 0, where 11 is any magnetic field. Show that if...for any closed surface S. (b) Determine the flux of a uniform magnetic field B throngh the curved surface of a right circular cone (radius R, height h) oriented so that B is normall to the base of the cone as shown in the figure.

Suppose that E is a one-dimensional normed linear space. Prove that E is complete and that each linear functional on E is continous.

Suppose that T is a topologocal vector space. Prove that a linear functional f on T is continuous if and only if ker(f) is closed.

Given that the acceleration vector is a(t) = (-9cos(-3t)) i + (-9sin(-3t)) j + (-2t) k , the initial velocity is v(0) = i + k , and the initial position vector is r(0) = i+j+k , compute: A. The velocity vector B. The position vector

Problem. Show that if is an orthonormal set in a Hilbert space H, then the set of all vectors of the form is a subspace of H. Hint: Take a Cauchy sequence , where . Set and show that is a Cauchy sequence in . Please see the attached file for full problem description.

An object of weight 20kN is supported from above by two cables inclined at 40 and 70 degrees to the horizontal as shown (please see attachment)

Please see the attached file for the fully formatted problems. Suppose that a mountain has the shape of an elliptic paraboloid , where a and c are constants, x and y are the east-west and north-south map coordinates and z is the altitude above the sea level (x,y,z are measured all in metres). At the point (1,1), in what direc