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Vector Calculus

Vector Projections

Let U and a be nonzero vectors Show explicitly that the the angle between: Proj a U = Projection of vector U along vector a and The vector U - Proj a U (vector component of U orthognal to a) is a right angle (i.e show that the dot product of these two vectors is 0)

Vector Spaces, Mappings and Kernels

Let V1 and V2 be finite dimensional vector space over K and let S Є L(v1, V2). Prove that there exists T Є L(V2, V1) such that S =STS and T= TST.

Writing Equations from Word Problems

Two planes left an airport at noon. one flew east at a certain speed and the other flew west at twice the speed. The planes were 2700 mi. apart in 3 hours. how fast was each plane flying? Write an equation and solve.

Vectors and Grad Proof

If V is a vector function, show the following by expansion that the following equality may be maintained: (V.grad)V=(gradxV)xV + grad(V^2/2) Please see the attached file for the fully formatted problems.

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)

Period of a vector function

I need assistance finding the period of a vector valued function. The function is of the form cos^2(2T)+Sin^2(3t) + cos(2t-pi/2) I remember doing something like this back in pre calc where the period was 2pi/b but the functions were not vectors. How would I proceed from here? Any help is appreciated.

Vertices in Tree

A tree has 11 vertices of degree 3, 12 vertices of degree 2, 10 vertices of degree 4 and the remaining vertices are of degree 1. How many vertices does it have?

Vector Spaces and Subspaces, Addition and Scalar Multiplication

Please view the attached files to see the expressions which are in question for parts A and B. 1. A) Determine whether the following sets are vector spaces, in each case giving reasons for your answer. B) Determine whether W is a subspace of the given vector space V.

Equilibrium vector

Find the equilibrium vector for each matrix M1 = .85 .15 M2= 3/5 2/5 .55 .45 1/4 3/4

Vector problem

Each square n*n region of an image yields a vector of length n^2 such that the components of the vector are the grey levels of the pixels in the square. Let u, v be the vectors obtained from two image patches, let a be the average of the entries in u, let b be the average of the entries in V and let e be the vector of length n^

Vector Fields

(See attached files for full problem description) For this one you need chapter 2 I think, it is problem number 4 page 57.

Tangent and Normal Unit Vectors

Please show a detailed solution to this problem. Pleas show the curve and vectors together in a sketch. Thank you! Find the unit vectors that are tangent and normal to the curve at the given point: y = e^x , (ln2,2)


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

Unit Normal Vector

Let vector r(t) = t i + t^2 j represent a plane curve. Find T(t), T(1) and N(1). Sketch the plane curve and graph the vectors T(1) and N(1) at the point t = 1.

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

Unit tangent and unit normal vectors.

Given the vector r(t) = t i + t^2 j find T(t), T(1) a dn then N (1). After this I am to sketch the plane curve and graph the vectors T(1) and N(1) at t = 1.

Unit Tangent Vector

I am asked to find the unit tangent vector at t=2 for the following : vector r(t) = t i + t^3 j + 3t k How do I do this problem and what is the final answer?

Evaluating the limit of a vector.

How do I evaluate the limit of the following vector: lim[(e^2t/t^2-1)i + (1-cost/t)j + sq rt (4-t^2)] t-->0 How do I go about solving this problem and what is the answer that will be obtained. I'm not even certain where I should begin.

Please explain in step by step detail the following

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 reac

Vector-Valued Functions

The problem asks me to sketch the curve represented by the vector-valued function. The vector-valued function is: r(theta) = cos theta i + 3 sin theta j The solution in the solution manual has the following: x = cos theta y = 3 sin theta Up to here I understand what is being done. T