4. Show that u and v are orthogonal if and only if ||u + v||2 = ||u||2 + ||v||2. (Please avoid an internal approach based on coordinates.)

Q: Given that the vectors X1 = () e- -2t ,X1 = (C)e-6t are solutions of a system x' = AX, do they form…

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

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

Q: How do you know that for all a, b, c, d: ax + bx will be orthogonal to cx? + d ?

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Q: Prove that if ||u|| = ||v||, then u + v and u- v are orthogonal.

A: Given ||u||=||v||

Q: (a) Prove that u + vand u - v are orthogonal in IR" if and only if ||u|| = ||v||. (b) Draw a diagram…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: D. Prove i [- 2,5, 1], 7 = [2,1, – 1] and w [- 6,0, – 12] are orthogonal. %3D

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Q: Find a vector v satisfying 3v + (5, 20) = (11, 17).

A: To find a vector v satisfying 3v+(5,20)=(11,17).

Q: Use Cramer's rule to solve for y without solving for the un- knowns x, z, and w. 4x + y+ z + w= 6 .…

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Q: 8. Let C = 2 c=(; 1) - (;) ()- and b = Find a 2-vector z such that Cz = b.

A: we have  C=112-1b=11   we have to find a 2-vector such that  Cz=b

Q: Let U(9, 9, 2), V(9, t, 4), and W(2, 5, 8). U, V and W have the same starting point. find t for…

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Q: (3,8, 0) is u x v = %3D So the cross product of u = (1, 0, 1) and v =

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Q: 4) Suppose x and y are vectors in R". Prove a) r+ >f -x->f =4(x• y) b) If x and y are orthogonal…

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Q: 11. Use vectors to show that the distance from P:(x1, yı) to the line ax + by = c is |ax, + by, – c|…

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Q: Show that u and v are parallel if and only if u×v = 0.

A: The cross product of two vectors are defined as u×v=uvsinθ. Two vectors are parallel if the angle…

Q: Prove that llu + v|| = llu - vll if and only if u and v are orthogonal.

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Q: Let ā = (1,2),b = (x,y) and = (1,-2) find the value of x and y such that a is %3D %3D %3D orthogonal…

A: Given that a is orthogonal to b So a·b=0also we have c·b=2

Q: _3. Let x be a convex combination of {y1, . . , yk }. Assume in turn that each y; is a convex…

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Q: Find u × v and show that it is orthogonal to both u and v.u = (−2, 1, 1), v = (4, 2, 0)

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Q: - Show that the vectors X, - () 5e3t and X2 = ) 2et 7et form a linearly independent set.

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Q: 2. Follow the directions of Exercise 1 using the weighted Euclidean inner product (u, v) = }u,V, +…

A: As per the company guidelines we are supposed to answer only first three subparts.Kindly repost the…

Q: Find u × v and show that it is orthogonal to both u and v.u = (2, −3, 1), v = (1, −2, 1)

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Q: Find the distance between

A: We use the distance formula to find the distance between u and v.

Q: Prove that for any vectors u, v, w , the set span{u+v,u – v, 3u – 3v + 2w} contains u, v, and w .…

A: A vector v is said to be contained in span of set S  (span(S)) if vector v can be written as a…

Q: 2. Use a directed line segment to represent the vector. (а) и %3D (2, —4) (b) v = (2,4, –4)

A: Given that the vector a u=2,-4  b v=2,4,-4Solution a u=2,-4u=22+-42u=20  =25b v=2,4,-4v=22+42+-42v=6

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: (2) Prove that if a + b and a – b are orthogonal, then a and b have the same magnitude.

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Q: Let w = -4xy – 5yz – 4xz, x = st, y = et, z = t Compute (3,4) = as (3,4)%3D at

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Q: If the vector u = 2i-3j+v3k , then the length of u is -v2 True O False

A: # we are entitled to solve one question at a time , please resubmit the other question if you wish…

Q: find the projection of v onto u.

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Q: a) Prove that ||u + v|| = ||u - v|| if and only if u and v are orthogonal, (b) Draw a diagram…

A: a) Suppose u,v are orthogonal. Now…

Q: Compute the component of u along v. u =〈3, 2, 1〉 v =〈1, 0, 1〉

A: Given the vectors are u=<3,2,1> and <1,0,1>.Our Aim is to compute the component of u…

Q: 21. Apply the conjugate-gradient method to the problem Ax = b, where 1 A = 1 1 %3D and b = 1 Verify…

A:

Q: 13. Recall that nonzero vectors u and are called collinear if there exists = a nonzero scalar a such…

A: Given data:- Nonzero vectors are u→ & v→ Nonzero scalar α v→=αu→

Q: Find u × v and show that it is orthogonal to both u and v.u = (4, 1, 0), v = (3, 2, −2)

A: Now cross product is,

Q: 8) Find the cross product a x b and verify that it is orthogonal to both a and b. a = (1, 1, -1),…

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Q: Compute the orthogonal projection of v = 3 onto the line L through -2 and the origin. projr (7) =

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Q: 19 - By using the definition of the dot product, which vector is orthogonal (perpendicular) to å =…

A: Dot product of two vectors

Q: Let u = (1, 2, 3, 4), v = (0, 2, t, 3), and w = (1, 1, 0, 1). Find t so that u + v and w are…

A: Here, u=(1,2,3,4),v=(02,t,3) & w=(1,1,0,1)

Q: 18) (Section 9.7) Find the cross product in space u = 3i – 2 j+4k v=i-3j-k

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Q: Find all values of c so that (2, 1, c) and (c – 1, 2, 2c) orthogonal. are

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Q: 2 Given u -1 and U , for which w is the set {u, v, w} linearly independent?

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Q: Show that u and v are orthogonal if and only if ||u + v||² = ||u||2 + ||v||². (Please avoid an…

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Q: 2. Show that the ry plane W = (r, y, 0) in R³ is generated by: (i)u = |2 and v= (ii)u = and v=

A: Please see the below picture for detailed solution.

Q: Suppose that u and v are vectors in 3-space such that |u||= 5 and u · v = 7.

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Q: If an unknown vector X satisfies the relations X·b = β and X×b = c, express X in termsof β, b, and…

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Q: Is it true that if x is orthogonal to v and w, then x is also orthogonal to v − w?

A: x is orthogonal to v and w. Then their dot product is zero. That is, ⇒x·v=0  &  x·w=0

Q: Suppose that u, u and w are vectors in a real inner product space such that (u, w) = 16, (v, w) = 5,…

A: Instead of u¯,v¯,w¯ , we are writing u,v,w. (for sake of convenience). Given: (i) <u,w>=16(ii)…

Q: Find u × v and show that it is orthogonal to both u and v.u = (12, −3, 1), v = (−2, 5, 1)

A: Given vectors u=12,-3,1v=-2,5,1 Their cross product is u×v=ijk12-31-251…

Q: In Exercises 1–5, find the length and direction (when defined) of u * v and v * u. 1. u = 2i - 2j -…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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.