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Vector Calculus

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

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)


1. Find the component form of the vector representing the velocity of a boat traveling at 8 knots, or nautical miles per hour, with a bearing of N 53-degrees W. 2. A force of 703 pounds is needed to push a stalled car up a hill inclined at an angle of 16-degrees to the horizontal. Find the weight of the car. Ignore fric

Vector-valued function word problem

A baseball is hit from a height of 3 feet with an initial speed of 120 feet per second at an angle of 30 degrees above the horizon. Find the vector-valued function describing the position of the ball t seconds after it is hit. To be a home run, the ball must clear a wall 385 feet away and 6 feet tall. Determine if this is a home

Problems Related to Vectos: Force, Motion, and Currents

Please explain in step by step detail the following: 25. A weight of 850 pounds is suspended by two cables. One cable makes an angle of 66 degrees with a vertical line, the other makes an angle of 42 degrees with a vertical line. Find the amount of force exerted by each of the cables. 24. An airplane is scheduled to reach

Vectors in spherical and cylindrical

(a) Given A = a*p_hat + b*psi_hat + c*z_hat (cylindrical unit vectors), where a, b, and c are constants. Is A a constant vector (uniform vector field)? If not, find: the divergence and curl of A (b) If A = a*r_hat + b*theta_hat + c*phi_hat in spherical coordinates, with constant coefficients. Is A a constant vector (unifor

Vector Spaces

How to prove or counter with example the following statements: (1) If two subspaces are orthogonal, then they are independent. (2) If two subspaces are independent, then they are orthogonal. I know that a vector v element of V is orthogonal to a subspace W element V if v is orthogonal to every w element W. Two subspaces W1

Proof of Vertex, Extreme Point, Basic Feasible Solution

Can you please let me know how to approach those proof questions. Consider the polyhedron P = {x &#61646; 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


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

Vector Field, Gradients, Div, Divergence, Curl and Surface Integrals

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

Potential and Electric Field Vector of Two Concentric Charged Cylinders

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.

Vector Calculus : Flux and Gauss's Law

(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.

Velocity/position vectors

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

Gradients : Elliptic Paraboloid and Vector Fields

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

Angle and Force Must the Second Tractor be Doing

1) A river 35 m wide flows south at a speed if 15 m/s. What must be the velocity and heading of the boat if it is to move directly from the west bank to the east bank in 5 seconds? 2) Two tractors are hooked to a combine. the combine needs to be pulled due east at 400 N. One tractor is pulling at 190 N, 32degrees S of

Vectors: Resultant Force

Three horses exert forces on a hitching pole. What is the resultant force on the pole, if horse A exerts a force of 350 N at an angle of 48 degrees N of East, horse B exerts a force of 560 N at an angle of 37 degrees N of West, and horse C exerts a force of 200 N at an angle of 87 degrees N of East?

What is the resultant force?

A force of 25 N acts perpendicular to another force of 22 N. If the forces act together on the same object, what is the resultant force?

Trees: Vertex; Cycle

Let G be a graph in which every vertex has degree 2. Is G necessarily a cycle? *Please see attachment for additional information. Thanks. Use words to describe solution process. Use math symbol editor like LateX, please no stuff like <=.

Vector Space Subsets and Subspaces

Let [a,b] be an interval in {see attachment}. Recall that the set of functions {see attachment} is a vector space over {see attachment} with addition (f+g)(x):=f(x)+g(x) and scalar multiplication a) choose [a,b]=[0,1]. Decide for each of the following subsets if it is a subspace. Justify your answer by giving a proof or a c

Vector projection and follow-up

The diagram attached shows a rectangular solid, two of whose vertices are A=(0,0,0) and G=(4,6,3). a) Find vector projections of AG onto the following vectors: AB, [0,1,0], [-1,0,0] and [0,0,1]. b) Find the point on AC that is closest to the midpoint of GH (The diagram is on page 21 and it is problem 6) (As you can see


Please see attachment. Require problems solving, also explanations etc for better understanding of vectors. VECTOR PROBLEMS (1) Let l be the line with equation v = a + t u. Show that the shortest distance from the origin to l can be written | a × u |