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Vector Calculus

Vectors in R^n, Orthogonal Spaces and Lines of Best Fit

1) Let u and v be vectors in R^n. a) Prove that llull = llvll if and only if u + v and u - v are orthogonal. b)Let (proj of u onto v) be the vector projection of u onto v. For u, v does not equal to 0, prove that (projection of u onto v - u) is orthogonal. 2) Find a basis for the space orthogonal to [1,1,0]^T in R^3.

Mechanics: Scalar and Vectors

1. Given the following 3 vectors, all of which lie in the horizontal plane, (see attachment for list of vectors), find: (a) 3A-B (b) 1) A?B 2) B?A (c) 1) A*B 2) B*A (d) (5A-6B+4C)?(B*C-A*B) *(Please see attachment for complete question and equations)

Vector Space Subsets and Subspaces

Let [a,b] be an interval in {see attachment}. Recall that the set of functions {see attachment} is a vector space over {see attachment} with addition (f+g)(x):=f(x)+g(x) and scalar multiplication a) choose [a,b]=[0,1]. Decide for each of the following subsets if it is a subspace. Justify your answer by giving a proof or a c

Properties of vector spaces.

Which of the following are subspaces of the vector space ? Justify your answer. A vector space in R^3 such that every vector (a,b,c) has the property: a-b-c=2 A vector space in R^3 which has the form (a,b,a+b)

Vector Subspaces Definition

If U is a subspace of V then W=V-U (a vector x that belongs to W can not belong to U) W also is a subspace. (Proof or counterexample)

Determine a Normal Vector to Surface at the Given Point

Let X: R^2 → R^3 be the parameterized surface give by X(s,t) = (s^2 - t^2 , s + t, s^2 + 3t) A) Determine a normal vector to this surface at the point (3, 1, 1) = X(2, -1) b) Find an equation for the plane tangent to this surface at the point (3, 1, 1).

Sketch/Drawing: Vectors

Could you please give me some sort of a sketch or drawing of what a set S (its interior and its closure) would look like when: S= {(x,y) : -1<= y < cosx, -2# < x <= 2#} Note: <= is less than or equal to < is less than # is pi (3.14....).

Vector Subspaces Explained

What would a vector v in R4 such that: V(1,2,1,0) T V(1,0,-1,1) T V(0,2,0,-1) = <v> AND find scalars a,b,c,d such that <(1,2,1,0),(1,0,-1,1),(0,2,0,-1)> = <v> Please note: <v1,...,vk> denotes the vector subspace of Rn generated by the vectors v1,...,vk and that for scalars a1,...,an belonging to R, V(a1,...,an) =

Vector projection and follow-up

The diagram attached shows a rectangular solid, two of whose vertices are A=(0,0,0) and G=(4,6,3). a) Find vector projections of AG onto the following vectors: AB, [0,1,0], [-1,0,0] and [0,0,1]. b) Find the point on AC that is closest to the midpoint of GH (The diagram is on page 21 and it is problem 6) (As you can see

Vector : Gradient

The temperature of a plate at the point (x,y) is given by T(x,y) = 300+ 3x^2 -2y^2. A heat hating ant is located at the point (3,2). In which direction will the ant begin to walk? Give a unit vector in that direction.

Multivariable Calculus : Double Integral - Polar Coordinate

( ∫ ^n_r means that n is on the top of the ∫ and r is on the bottom) Evaluate the given integral by first converting to polar coordinates: ∫ ^2_1 ∫ ^(square root of 2x - x^2)_0 (1/(square root of x^2 + y^2)) dy dx ∫: is the integral symbol

Multivariable Calculus

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you. ( &#61682; ^n_r means that n is on the top of the &#61682; and r is on the bottom) Evaluate the iterated integral: &#61682; ^2_-1 &#61682; ^3_1 (2 x - 7y) dy dx &#61682;: denotes an integral

Type of Vectors in Matrix

The set of vectors {[ 1 -1] , [ 1 -1] , [ 2 -1] } [ 2 0 ] [ -1 0] [ -1 0] from M_2(R) is: A. linearly dependent B. linearly independent C. orthogonal D. a spanning set for M_2(R) E. a basis for M_2(R)

Vectors

Please see attachment. Require problems solving, also explanations etc for better understanding of vectors. VECTOR PROBLEMS (1) Let l be the line with equation v = a + t u. Show that the shortest distance from the origin to l can be written | a × u |

Vector Operations Direction Angles

Given vector v with /v/ = 4 and direction angle of 45 degrees, write v in the form <a,b> a= 4cos 45 degrees= 4*.707=2.83 b= 4sin 45 degrees= 4*.707=2.83 vector v (a,b) becomes v(2.83, 2.83) compute (2*w).(u-v) where w = <-1, 0> 2(2<100)(5<60-4<45) 2*2<180*1.536<102.5 =6.144<282.5 or 6.144<-77.5degrees vector from equat

Normal Gradient Vector

Use the normal gradient vector to write an equation of the line (or plane) tangent to the given curve (or surface) at the given point P: x^(1/3) + y^(1/3) + z^(1/3) = 1; P(1, -1, 1).

Vector and direction angle

Please show me the steps Tahnk You A. write u in the form <a , b> B.Compute (2*u).v, where v=<sqrt(3) , -3>

Vectors : Dot Product

Let u=<-2,1> v=<3,4> w=<-5,12> Use properties of the dot product. u(Dot)(v-w) or u.(v-w)

Moorean 3-Space

Please see the attached file for full problem description. 1. Demonstrate (check the properties) that the following function is an inner product in R^3. (Call R^3 with this inner product Moorean 3-space). Let u=(u_1,u_2,u_3) and v=(v_1,v_2,v_3). Then <u, v> = uAv^T, where A=[ 2 0 0 ]

Vector Spaces : Rank

Please see the attached file for the full problem description. 1. Find the rank of A= [1 0 2 0] [ 4 0 3 0] [ 5 0 -1 0] [ 2 -3 1 1] . Show work. Help:

Vectors : Planes, Points, Cross Product and Dot Product

(1) a. Find the (vector) equation of the plane passing through the points (1,2,-2), (-1,1,-9), (2,-2,-12). b. Find the (vector) equation of the plane containing (1,2,-1) and perpendicular to (3,-1,2). (2) Suppose a, b, c are non zero vectors. a. Explain why (a x b) x (a x c)

Linear Alegbra : Vectors

Find equations for the indictated geometrical objects The line through the point P=(1,1,1) and perpendicualar to the plane 4x-2y+6z=3

Linear Alegbra : Vector Space

Let V= (x,y) in R2{y=3x+1} with addition and multiplication by a scalar defined on V by: (x,y)+ (x',y')= (x+x',y+y'-1) k(x,y)=(kx,k(y-1)+1) Given that with these definitions, V satisfies vector space axioms 1,2,3,6,8,9,and 10 determine whether or not V is a vector space by checking to see if axioms 4,5,7,are also satisfied.

Vector space and basis

Consider the following elements of the vector space P3 of all polynomials of degree less than or equal to 3. p(x)= x-1, q(x)=x+x2, r(x)= 1+x2-x3 Do these three polynomials form a basis for P3?

Vector analysis and Stroke's theorem

Apply Stroke's theorem to evaluate the integral over C of (ydx + zdy + xdz), where C is the curve of intersection of the unit sphere x^2+y^2+z^2=1 and the plane x+y+z=0, traced anticlockwise viewed from the side of the positive x-axis