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# Vector Calculus

### 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

### Hilbert Space and Subspace

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.

### Find graphically the tensions in the cables

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)

### 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

### Find a unit vector perpendicular to a surface.

Please see the attached file for the fully formatted problems. Find a unit vector perpendicular (normal) to the surface S given by z = (x^3)(y^2) + y + 4.

### Parametric Representation of Acceleration and Velocity

Please see the attached file for the fully formatted problems. Show that, if the acceleration of an object is always perpendicular to the velocity, then the speed of the object is constant. (hint, the speed is given by ). Show that, at a local maximum or minimum is perpendicular to .

### Vector Functions : Parametric Representations of Curves

Please see the attached file for the fully formatted problems. 1. What kinds of curves are given by the following parametric representations? ? ? ? 2. Find a parametric representation of each of the following curves: ? ?

### Vectors : Work Done on a Box Pushed up a Ramp

A woman exerts a horizontal force of 10 pounds on a box as she pushes it up a ramp that is 7 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box.

### Area of Parallelogram Given Vertices

Find the area of the parallelogram with vertices (3,4), (5, 7), (9, 11), and (11, 14).

### Vectors : Finding Work from A Force Vector

A constant force F= -3i + 8j +10k moves an object along a straight line from point (-6, -5, -4) to point (-3, 1, -6). Find the work done if the distance is measured in meters and the magnitude of the force is measured in Newtons.

### Scalar and Vector Products

We have three vectors: A = i + 2j ; B = 2j + k ; C = 2i + k a. Find the scalar triple product by direct multiplication. and b. The vector triple product by direct multiplication.

### 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?

### Finding Resultant Force of an Object: Example Problem

What is the resultant force of an object if there are forces of 70 lbs at an angle 90 degrees N of west, 55lbs at an angle of 63 degrees S of west, and 42 lbs at an angle of 158 degrees N of east?

### Resultant Force Vector Calculations

A chain is wrapped around a log and forces of 367 and 483 N are exerted at right angles to each other. What is the resultant force?

### What is the effective force? How far is the displacement

1) What is the effective force acting to push a 57lb object up a 35 degree slope? 2) A person walks 450 feet at an angle of 53 degrees N of W. How far is the displacement west and how far north?

### Horizontal and Vertical Components on Forces

FIND THE HORIZONTAL AND VERTICLE COMPONENTS OF THE FOLLOWING FORCES:(237LBS, 48 DEGREES),(369LBS,248 DEGREES) ALSO FIND THE X AND Y COMPONENTS OF THE VECTOR (529m,342DEGREES)

### VECTOR: Resultant Displacement

A person walks 18m East and then 32m in a direction of 65 degrees N of E. What is the resultant displacement?

### 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?

### Vectors and Resultant Force

What is the resultant force for the following forces? 250 N due north, 525 N due south and 238 N 38 degrees south of west.

### Vector: Force Magnitudes

A person pushes with a force directed along the lawn mower handle, which makes an angle of 52 degrees with the ground. What must be the magnitude of the person's force in order to produce a horizontal force of 35 lbs?

### Determining Resultant Force: Example Problem

A rope is wrapped around a pole so that a force of 75 lbs acts on one end and a force of 53 lbs acts on the other end. If the angle between the two forces is 114 degrees what is the resultant force? What angle does the resultant force make with the 53 lbs force?

### Vectors and Force of a Docked Boat

Find the force on the docking cable of a boat on which the wind acts in a northerly direction with a force of 235 lbs and the tide acts in an easterly direction with a force of 323 lbs.

### Vectors : Components of Force Vector

WHAT ARE THE X AND Y COMPONENTS OF A FORCE OF 185 N AT 132 DEGREES? WHAT ARE THE X AND Y COMPONENTS OF A VECTOR OF 432LBS, 24 DEGREES S OF E.

### Vectors for Horizontal and Vertical Components

A block of wood weighing 35 lbs is resting on an inclined plane sloped at 36 degrees to the floor. What is the component of weight down the plane? What is the component of weight perpendicular to the plane?

### Vector Components : Weight

A 850 LB PULL ACTS IN A DIRECTION 32 DEGREES S OF E. WHAT IS THE EASTWARD COMPONENT? WHAT IS THE SOUTHWARD COMPONENT?

### Equivalence of Versions of Angular Momentum Equations

I need to show that the following two terms are equivalent: l = m(r2I - rr)xΩ l = r x mv = r x m(Ωx r) where r is the position vector from the origin to the particle l is the angular momentum I is the identity tensor Ω is the vector angular velocity x indicates a cross product rr is a dyadic product.

### Vector Fields for Evaluating Integrals

Let a vector field F be given by F(x,y,z) = (x^3)i - (y^2)j + (2yz)k and a curve C be given by r(t) = 2ti + sintj - costk, 0 <= t <= (pi/2) 1. Evaluate the line integral F*dr. 2. Determine the arclength variable s from t. 3. Determine the unit tangent vector T(s). 4. Evaluate the total arclength L. 5. Write th

### 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 <=.

### Strictly Separating Set Characteristics

For both 1 and 2, could you tell me whether or not there is a hyperplane that strictly separates the given sets A,B. If there is, find one. If there is not, prove so please. 1) A={(x,y):abs(x) + abs(y) <=1}, B={(1,1)} 2) A={(x,y):xy >= 4}, B={(x,y):x^2+y^2 <= 1} where abs = absolute value
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