Let W be the union of the first and third quadrants in the xy-plane. That is, let х :xy a. If u is in W and c is any scalar, is cu in W ? Why? b. Find specific vectors u and v in W such that u + v is not in W. This is enough to show that W is not a vector space.

Q: Let V be the set of vectors shown below. V= x> 0, y>0 a. If u and v are in V, is u +v in V? Why? b.…

A: Please check step 2 for solution.

Q: let ej denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Prove that…

A:

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B'A) Given the…

A:

Q: Find a unit vector normal to the plane containing v=2i+3j-k and . w = -2i - 4j - 3k.

A: Given: v=2i+3j-k  w = -2i - 4j - 3k.

Q: Find the unit normal vector on the following surface z = √√x² + y²

A: Please find attachment below

Q: Consider (u, v) = 0 for any vectors u, v e R4. Is this an inner product? Explain why or why not.

A:

Q: Find a formula for the projection of any given vector w onto the plane u · x = 0 in R', where u is a…

A: The objective is to find the formula for the projection of any given vector w onto the plane u.x=o…

Q: Let II be a plane passing through the origin, and let n be a non-zero vector orthogonal to II. Then…

A:

Q: The curd of the vector funchion v = Jx i+ Jgj+ ZK is ☺ xi+ yj+ zK

A: It is given that, v→=x i^+y j^+z k^   According to curl, the swirl of a vector around a point. It is…

Q: If w is a vector such that w+(ŵ×ü) = v,then prove that only if u is perpendicular to v.

A:

Q: Let u be a vector in R² whose projection onto the x-axis is shown in the figure. Determine the…

A: Given:

Q: Given a vector x € RN and let y = Hx be the reflection of x through a hyperplane L, where H = I–…

A: Asp per the question we are given a reflection matrix H along a hyperplane with unit normal vector u…

Q: Let t e R. Discuss the lincar independence of the vectors over R: aj %3D(1, 1, 0), аз %3 (1, 3, —1),…

A: we will use the connection of linear independence of a set and columns of an Invertible mat

Q: Consider (u, v) = 0 for any vectors u, v e R4. Is this an inner product? Explain why or why not. %3D

A:

Q: Let f : R² → R be defined by S if (x, y) ± (0,0), 1³y? f(x,y) = if (x, y) = (0,0). Determine the…

A: According to the given information, it is required to determine a non-zero vector u = (h,k) for…

Q: Let V be the set of vectors shown below. V= x<0, y< a. If u and v are in V, is u + v in V? Why? b.…

A:

Q: Consider the vector v = 1 and the vector "function" w(k) = 1 For what value of k are i and w(k)…

A: Two vectors v→ and w→ are said to be orthogonal if v→·w→=0 Given v→=111w→=11k

Q: Use the vector forms of Green's Theorem to calculate (a) F•n ds and (b) F:T ds for F=y°i+x°j, where…

A:

Q: Find the net outward flux of F = a x r across any smoothclosed surface in ℝ3, where a is a constant…

A: Net outward flux is ∇.F ∇=i∂∂x+j∂∂y+k∂∂za=a1i+a2j+a3kr=xi+yj+zk

Q: Let f(x, y) = x^2/(x + y). Find a unit vector u such that Duf(1, 2) = 0

A: f(x, y) = x^2/(x + y)

Q: Let ƒ : R² → R be defined by A if (x, y) # (0,0), f(r, y) = if (r, y) = (0,0). Determine the nonzero…

A: According to the given information, it is required to determine a non-zero vector u = (h,k) for…

Q: Let H be the set of all points in the third quadrant in the plane V = R². That is, H = {(x, y) | x <…

A:

Q: Consider the real vector space M2x3(R) with the usual inner product (А, В) — tr (B'A) Given the…

A:

Q: Find the spacing and involution for the following vector functions 1- ƒ(x, y, z) = (xi + yj + zk]…

A: Topic:- vector

Q: Consider the vector space C2 with the standard inner product. Define the linear operator T:C? → C2…

A:

Q: Let P be the plane in space that intersects the x-axis at 5, the y-axis at −5, and the z-axis at 4.…

A: Given query is to find the perpendicular vector to the plane p.

Q: Let, T:R' → R;T(x, y, z) = (x + y+ z,2y + z,2 y+3z) verify that [T][v]s = [T (v)]; for any vector v…

A:

Q: Let V be an n-dimensional vector space. Define f : V→R by f(v) =|v|². Let p, qe V. Find d fp(a).

A:

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B'A) Given the…

A:

Q: Find x such that the vectors xi+2j -3k, i+j ,j+3k are Coplanars (lies the Same Plane) . on

A:

Q: Classify the following vector fields H = (y+z)i+(x + z)j+ (x + y)k, O(a) solenoidal O(b)…

A: The vector field H=(y+z)i+(x+z)j+(x+y)k, we have to classify this.

Q: Let, T :R → R';T(x,y,z) =(x+y+ z,2y+ z,2y +3z) 7. verify that [T][v]s = [T (v)], for any vector v in…

A:

Q: Compute the curl of the following vector V defined by V(x,y,z) = e(x²+y² +z²) y

A:

Q: Consider the set V of all vectors of the form (x y 2) where x, y and z are real numbers 3. Show that…

A:

Q: In the inner product space R with the dot product, the orthogonal projection of the vector 1 onto…

A: We have to find the orthogonal projection of the given vector.

Q: Let u= and w- Find the unique vector v in R'such that u.v 1, v is orthogonal to w, l|v|| V3, and y>…

A:

Q: Show that the vector V = (x+ 3y)i + (y – 32) j+(x – 2z)k is solenoidal. %3D

A: Solenoidal Vector : A vector function whose divergence vanished identically is called a solenoidal…

Q: Let II be a plane passing through the origin, and let n be a non-zero vector orthogonal to II. Then…

A:

Q: Find the vector projection of vector u onto vector v, and find the vector component of u orthogonal…

A: The projection of vector u onto v is given by projvu=u⇀v⇀v⇀2v⇀

Q: Let V = R?, defined the addition by: %3D (u1 +u2) + (v1 + v2)= (U1 +V1 , U2 +V2 + 1) With standard…

A:

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B' A) Given the…

A:

Q: Let p = 2 In(r), where r = V x2 + y2. Express Vo in terms of the unit radial vector e, in R2.

A:

Q: Let u= 2 and v= 2. Show that among all the scalar multiples cv of the vector v, the projection of u…

A:

Q: FIND THE FLUX F OF THROUGH 9, SSF. Nds, WHEREN IS THE UPWARD UNIT NORMSAL VECTOR TU S. Xi+ YJ 2 x² -…

A: Introduction: Double integral is applicable to determine the flux of a vector field through a given…

Q: Let II be a plane passing through the origin, and let n be a non-zero vector orthogonal to II. Then…

A: 3 dimensions plane questions

Q: If x = (x1, x2)T , y = (y1, y2)T, and z = (z1, z2)T arearbitrary vectors in R2, prove that xTy = yTx

A: x = (x1, x2)T , y = (y1, y2)T are arbitrary vectors in R2 .

Q: Let, T:R' → R';T(x, y,z) = (x – y,2x + 3y + 2z,x+ y+2z) . verify that [T][v], = [T (v)]; for any…

A:

Q: Let W = {[x]∈ R2^ | y > or equal to 0} .             [y] (a) If u and v are vectors in W , is u + v…

A:

Q: Let f : R" → R" be a function with the property that for any v, w E R" it holds that f(v) · f(w) = v…

A: Given the function f:ℝn→ℝn. Given that, this function preserves the operation 'inner product'. Which…

Q: Let g(x, y, z) = x²yz?. Find a (three-dimensional) unit vector u which satisfies Dug (1, 1, 1) = 0.

A: