### 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)

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)

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?

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. See attached file for full problem description.

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.

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

Find the equation of a plane through the origin and perpendicular to: x-y+z=5 and 2x+y-2z=7

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|

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) .

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

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

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

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)

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

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,

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

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

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

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⊕We. (ii) Find a pro

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

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.

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

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

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

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 ∂Q/∂x - ∂P/∂y. Find the scalar curl of F(x, y) = sin (xy) i +

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

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.

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

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

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