Show that (x, y) = 4x1y1-7x2Y2 for vectors x = (x1, x2) and y = (V1 Y2) in R2 defines an inner product.

Q: Find a unit norm vector that is orthogonal to both the vector (1,2, –1) and the line of intersection…

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Q: Find one voctor in R' that spans the intersoction of U and W where U is the xy-plane-that is, U =…

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Q: Find a linear mapping G that maps [0, 1]×[0, 1] to the parallelogram in the xy-plane spanned by the…

A: A linear mapping G that maps [0, 1]×[0, 1] to the parallelogram in the xy-plane spanned by the…

Q: Use Lagrange multipliers to find a vector in 3-space whose length is 3 /3 and whose components have…

A: In the equation ∇g=λ∇f, delta represents partial derivative. A partial derivative is a derivative in…

Q: An inner product (•, ·) on R³ under which the vectors 1 Vị = 1 1 , V2 = 2 satisfy (v1, v2) = 0…

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

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Q: Find a nonzero vector parallel to the line of intersection of the two planes 2y 5x - 3x = -5 and 5y…

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Q: Why are there no vectors u and v in R,, such that ||u.||= 2, ||v|| = 3, and    uxv= -7?

A: The given values are, u=2, v=3,   and  u×v=-7. The objective is to find the valid reason for the…

Q: Show that, in an inner product space, there cannot be unit vectors u and v with <u, v > <…

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Q: Find the cosine of the angle between the vectors P1(x) = 1+ 2x -x² and p2 (x) = -2-x+ 3x2 with…

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

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

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Q: Use Lagrange multipliers to find a vector in 3-space whose length is 7 V3 and whose components have…

A: By using Lagrange multipliers we have to find a vector in 3-space whose length is 7√3 and whose…

Q: The cross product of two vectors in R yields a scalar. True False

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Q: Use Lagrange multipliers to find a vector in 3-space whose length is 4 V3 and whose components have…

A: Given:     Let v=x,y,z be a vector in R3 with length v=43 This is x2+y2+z2=43            x2+y2+z2=48…

Q: Use the definition of the cross product to prove that the cross product of two parallel vectors is…

A: The cross product of two parallel vectors is zero. This can be proved by two definitions of cross…

Q: What is the definition of the triple scalar product for vectors u, v, and w in ℝ3?

A: To Define: Scalar Triple Product for vectors u, v and w in ℝ3.

Q: Find two vectors of norm 1 that are orthogonal to the vectors u = and v =

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Q: (B) Show that the area of triangle is - ÁxB (Using Vector method)

A: Let, a parallelogram be:

Q: Show that except in degenerate cases, (u * v) * w lies in the plane of u and v, whereas u * (v * w)…

A: We will prove that  (u*v)*w=(u.w)v-(v.w)u u*v=ijkuxuyuzvxvyvz…

Q: Find a vector v→parallel to the intersection of the planes 3x+y+2z=3 and       −(x+2y+4z)=7

A: Vector parallel to intersection of two planes is cross product of the normal vectors of the two…

Q: Use Lagrange multipliers to find a vector in 3-space whose length is 9/3 and whose components have…

A: We will use Lagrange Multiplier method to  maximize / minimize a function f(x,,y,z) subject to…

Q: Use Lagrange multipliers to find a vector in 3-space whose length is 6/3 and whose components have…

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Q: ^2

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Q: Defined byT (x, y, z) = (y+2z, –2x+y+2z, –3x– y+52), is diagonalizable.

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Q: Write the equation of the plane with normal vector n = 4k passing through the point (7, 2, 5) in the…

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Q: Let x and y be two orthogonal vectors. Then prove that +

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Q: Show that the point 1, - 2, 0 lies on both planes 3x-2y+z=7 and x-4y-2z=9. Hence, find the vector…

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

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Q: Show that the vector ñ = 2î – 3j is perpendicular (orthogonal) to the line 2x – 3y – 6 = 0.

A: to Show that the  given vector is perpendicular to the  given line

Q: Find a vector v parallel to the intersection of the planes 3x + y + 2z = 3 and – (x + 2y + 4z) = 7.

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Q: Can you find two vectors in R whose dot product is equal to their cross product? No, because the…

A: (iii) can you find two vectors in ℝ3whose dot product is equal to its cross product as we know , dot…

Q: Evaluate The Work Done By The Force Vector, F= (6x^2+3y)I – 15yzj + 10x^2zk From (0,0,0) To (1,1,1)…

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Q: Let p and q be two non-zero vectors. By using the definition of vector product show that pxq =…

A: In the given question, calculate the area of the parallelogram where two sides p and q are given.…

Q: Find the equation of the plane in xyz space through the point P= (3,3,4)and perpendicular to the…

A: We need to find equation of plane.

Q: uppose ā and b are two non-zero vectors in R3 and they are not parallel to each other. Then, their…

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Q: . Let x and y be two orthogonal vectors. Then prove that

A: If x is any vector then Euclidean norm or 2-norm is denoted by ||x||2 and is given by  ||x||2 =…

Q: Find a nonzero vector parallel to the line of intersection of the two planes 5y – 4x + z =-1 and –…

A: The line of intersection of two planes is orthogonal to their respective normals.

Q: Use Lagrange multipliers to find a vector in 3-space whose length is 4/3 and whose components have…

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Q: Is it always true that T(0) = 0 where the zero vectors live in the appropriate spaces? Justify

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Q: 2. Can three vectors in the xy plane R2, have u v < 0, v w < 0, w u<0?

A: Given that, u, v and w are vectors in XY plane. The dot product of vectors is defined as, if u = a1…

Q: Show with the help of the calculation rules for the scalar product for vectors ||x – 2 y|| = ||x||2…

A: The property used here are b2=b·b x·y=y·x cx·y=cx·y=x·cy Consider the left hand side of the given…

Q: 2) Find the vector from the origin to the center of the sphere 2x² + 2y² + 2z2 + 4x - 4z = 0.

A: The equation of the sphere is, 2x² + 2y² + 2z² + 4x - 4z = 0 So, x² + y² + z² + 2x - 2z = 0 So, (x²…

Q: In Rn with inner product{x, y} = xTyderive a formula for the distance between two vectorsx = (x1, .…

A: Take two vectors x and y in an inner product space x=x1,…,xn and y=y1,…,yn then x,y=xTy

Q: find two distinct vectors on the line belonging to the intersection of the hyperplanes: Xi+ 2x2 - X3…

A: We need to find two distinct vectors on the line belonging to the intersection of the hyperplanes

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: Suppose that two vectors T and ỹ in R" are orthogonal, meaning their inner product satisfies ·g = 0.…

A: This is a problem of vector calculus.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.