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# Vector Calculus

### The equation of a plane

Part I: <(u x v), u> is greater than 0 for all vectors u and v in R^3. Part II: The equation of the plane through the origin determined by two vectors u and v in R^3 is <(u x v), x> = 0. Are these statements true or false and please explain why.

### Vector calculations

Show that (a x b)â?¢(c x d)=(a â?¢ c)(b â?¢ d)-(a â?¢ d)(b â?¢ c). Suppose that A, B, C are three points equidistant from O and that the segments BC, CA, AB subtend angles x, y, z respectively at O. Show that cosx=cosycosz+ sinysinzcosw, where w is the angle between the planes AOB and AOC.

### Vector question

Solve the simultaneous vector equations for r: r x a=b, r â?¢ c=a, given that a â?¢ b=0 and a is not equal to 0. Distinguish between the cases a â?¢ c is not equal to 0 and a â?¢ c=0. Note: r x a is the cross product resulting in a vector b and r â?¢ c is the dot product resulting in a scalar a.

### Subsets of a normed vector space

1. If A and U are two subsets of a normed vector space, and U is open, show that A+U is open. Here A+U={a+u | a belongs to A and u belongs to U}. 2. Suppose {xn} from n=p to infinity is a non-converging subsequence in a compact set of a metric space. Show that it has two convergent subsequences which converge to distinct limi

### Identify vector spaces

Briefly explain, which of the following are vector spaces? (a) The set of all real symmetric 3x3 matrices. (b) The set of all real 3x3 non-singular matrices. (c) The set of all real functions f(x) which are continuous for all x.

### Stokes Theorem: Example Problem

Given the vector field F=3yi + (5-2x)j + ((z^2)-2)k find a) Div F b) Curl F c) The surface integral of the normal component of curl F over the open hemispherical surface (x^2)+(y^2)+(z^2)=4 above the xy plane. * Hint: by a double application of Stokes theorem, part c can be reduced to a triviality

### Vectors Characteristics of Initial Points

Suppose A(3,-1,0) and B(-4,-2,3) are 2 points in 3-space. Find a vector with the following three characteristics: initial point at the origin, collinear but in the opposite direction of vector AB , length 3

### Vectors

A= <1,2,3> b=<4,5,6> c=<7,8,0> A) Find a x b (cross product) B) Find the volume of the parallelpipe with adjacent edges a,b,and c. C)Determine if the vectors a and b are parallel, perpendicular or neither.

### Projecting a Vector onto W

Need help understanding how to do this problem: Let C^3 be equipped with the standard inner product and Let W be the subspace of C^3 that is spanned by u=(1,0,1) and u2=(1/sqrt3, 1/sqrt3, -1/sqrt 3). Project the vector v=(1,i ,i ) onto W.

### Open Subset Metric Space

Let (X_i, d_i) be metric spaces. Let X=X_1 x X_2 x .... x X_n and let (X,d) be the- metric space defined in the standard manner : d(x,y)=max(d_i(x_i,y_i)) . For i = 1,2,...n , let O_i be an open subset of X_i. Prove that the subset O_1 x O_2 x ... x O_n of X is open and that each subset of X is a union of sets of this form.

### Plane equation

A) Write a plane equation for plane passing through P (1,2,3) and perpendicular to n = <5, -3, 2>. b) Write a plane equation for plane passing through P(1, 2, 3), Q(-1,0,1) and R(0,0,1). Find the x, y, and z intercepts and sketch the plane.

### Normal and binormal vectors to a curve and normal plane

If C is the curve given parametrically by R(t)=cost(i)+sint(j)+2t(k) find a) the normal N and the binormal B for this curve @ t=0 b) the equation of the plane passing through the point R(0) and parallel to both vectors N and B of part(a)

### Vector Depicting Lengths

A 100g mass is placed at 20 degrees and a 200g mass at 120 degrees on a force table draw the vector diagram to scale using 0.2N/cm and calculate the resultant vector magnitude and direction graphically using the parallelogram method. Then calculate the resultant using the analytical methods of adding vectors.

### Decomposition of a vector.

Decompose 6i-3j-6k into vectors parallel and perpendicular to following vectors. a) i+j+k b) 2i-j-2k c) 2j-k

### Component form of vector...

Find the component form of the vector v that has an initial point at (1,-2,2) and a terminal point at (3,-3,0) find the component form of the vector w if ||w||=4 and points in the direction opposite v. hint: normalize v first.

### Angles between Planes with Vectors

1. Find the angle between the planes with the given equations. 2x - y + z = 5 and x + y - z = 1 2. Find the values of r' (t) and r'' (t) for the given values of t. r (t) = i cos t + j sin t; t = pi/4 3. The acceleration vector a (t), the initial position r = r (0), and the initial veloc

### Perpendicular Vector Planes Through Points

1. Two vectors are parallel provided that one is a scalar multiple of the other. Determine whether the vectors a and b are parallel, perpendicular, or neither. a = 12i - 20j + 17k and b = -9i + 15j + 24k 2. Find a unit vector n perpendicular to the plane through the points P(1, 3, -2), Q(2, 4, 5), and R(-3

### Translation and reflection

A student claims that anything that can be accomplished by a translation can be accomplished by a reflection. She claims that if A' is the image of A under a translation, then A' can be obtained by a reflection in the line(which is the perpendicular bisector of AA'. Hence, a translation and a reflection are the same. How do you

### linear combination of the given vectors

Suppose {v_1, v_2, v_3} is linearly independent set of vectors in R^n. Determine which of the following sets of vectors are linearly independent and which are linearly dependent. If a set is linearly dependent give, a linear dependence relation. (Use the following technique. If {w_1, w_2, w_3} denotes one of the sets below, solv

### linearly dependent set of three vectors

What is an example of a linearly dependent set of three vectors with the property that any single vector can be removed from the set without changing the span of the set? What is an example of a linearly dependent set of three vectors that does not have the property as Question above?

### Vector Calculus: Example Problems

Let p1 = (4,0,4), p2 = (2,-1,8), and p3 = (1,2,3). a) Show that the three points define a right triangle. Hint the difference between two vertices is a vector whose direction coincides with that of a triangle side, and a pair of such vectors must be orthogonal in order for the triangle to be a right triangle. b) Specify

### Vectors: Resultant Vector

1. Two students are using ropes to pull on a heavy object, as shown in the diagram below. a. Using your knowledge about right triangles, with how much force will the object move? (2 marks) < > b. Solve for the angle of the object relative to the 500 N force, to the nearest degree. Use your knowledge about right triang

### Vector Calculus Proof: Example Problem

The midpoint of a side of a triangle in R^3 is the point that bisects that side (i.e., that divides it into two equal pieces). Let triangle in R^3 have sides A,B and C and let denote L denote the line segment between the midpoints of A and B. Prove that L is parallel to C and that the length of L is one-half the length of C.

### Divergence and Curl of a Vector Field

I need all work shown and step-by-step solution. 1. Consider the vector field v(x,y,z)=(x2,-yx,z-xz). (a) Compute the divergence and curl of v. (b) Show that v is neither the gradient of a function nor the curl of a C2 vector field.

### To find inner product in vector spaces

Find each of the following for the given inner product defined in R^2 a) d(u,v) b) < u,v > < u,v >=3u(subscript 1)v(subscript 1) + u(subscript 2)v(subscript 2) where u=(-4,9) and v=(0,4)

### Component Form of Vectors

See attached file. Find the component form of the specified vector given that u=....

### Vector Calculus: Principle Normal Vector and Binomial Vector

Please see the attached file for the fully formatted problems. If T'(t) does not equal 0, it follows that N(t) = T'(t)/||T'(t)||is normal to t(t); is called the principle normal vector. Let a third unit vector that is perpendicular to both T and N be defined by B = T x N; B is called the binomial vector. Together, T, N and B

### Vector Calculus : Vector Valued Functions, Arc Length and Curvature

Please see the attached document regarding homework specifics. The first page is the exercise and the second page is the solution from back of the book. Thank-you so much for your help.

### Gradient Vector and Tangentl Line

If g(x,y)= x-y^2, find the gradient vector (3,-1) and use it to find the tangent line to the level curve g(x,y)= 2 at the point (3,-1). Sketch the level curve, the tangent line, and the gradient vector.

### Axis of Symmetry Vertex

Write an equation for axis of symmetry; find coordinates of vertex y = 3x squared + 21x - 4