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# Vector Calculus

### Vector Proofs : Dot Products and Cross Products

Show that (A x B).(C x D) = (A.C)(B.D) - (A.D)(B.C) Show that [A x (B x C)].B = (A x B).(A x B)

### 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.

### Vertices, Asymptotes of a Hyperbola & Polar Coordiates

Find the vertices of the ellipse Xsquare/4 +Ysquare/25=1 Find equations for the asymptotes of the hyperbola X square/9 - Y square/81 = 1 Change the polar coordinates of (5,pi) to rectangular coordinates.

### Resultant Displacement for Two Vectors

Question: Suppose that you travel north for 65 kilometers then travel east 75 kilometers. How far are you from your starting point?

### 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|

### Mathematics - Calculus III - Equation of Tangent

Find the equation of the tangent to the surface at the indicated point: x = u^(2), y = v^(2), z = u + v ; (0, 2, 0) .

### Finding Vertex and Intercepts

F(x) = x^2 + x - 2

### 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)

### Vectors and Normal Components

^ ^ ^ ^ ^ For v = -4 j , a = 2 i + 3 j find a (subscript)T and a (subscript)N.

### 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

### Domain of a Vector Function

^ Find the domain of r(t) = <2e^(-t), sin^(-1) t, ln (1 - t) >; t(subscript 0) = 0

### 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 Fields : Divergence and Curl

Calculate the divergence and curl of the vector field F(x,y,z) = 2xi + 3yj +4zk.

### 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

### Domains of Vector Functions

See attached file for full problem description. Need problems 1,2,3 from the file...exercise 13.1.

### 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 +

### Vertex / Focus of a Parabola

Find the vertex and the focus of the parabola y=x2 + 10x + 22

### Vectors : Is the Quadrilateral a Square?

Show that the quadrilateral with vertices A = (-3, 5, 6), B = (1, -5, 7), C = (8, -3, -1), D = (4, 7,-2) is a square.

### 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