Prove that two vectors v = (a, b) and w = (c, d) are perpendicular if and only if ac + bd = 0

Q: Prove Ảx (B x Č) = B(Ả C) – Č(Ã B) is true by expanding Ả, B, and components (or Cartesian…

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Q: Prove that for any non-zero vectors a, b, a · b = |a||b| cos(0), where 0 is an angel between a and…

A: Consider the n -dimensional vectors a and b , where n≥2 . The vectors a , b and a−b can be…

Q: Compute the angle between the vectors u = [2, 1, -2] and v = [1, 1, 1].

A: To find out the angle between two vector use the formula. First find dot product of these two…

Q: Prove the stated property of distance between vectors. d(u, v) = d(v, u) for all vectors u and v

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Q: Prove the stated property of distance between vectors.d(u, v) = d(v, u) for all vectors u and v

A: Given the stated property d(u, v) = d(v, u). We need to prove the above stated property.

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: 2. Show that if the vectors v,, v2, .,Vm are mutually orthogonal then they are linearly ...…

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Q: let v = (1, 3, −2) and w = (2, −1, 4). Find all the vectors orthogonal to both v and w.

A: Given: The vectors v=1,3,-2 and w=2,-1,4. To find: The vectors orthogonal to both v and w.

Q: u.v<0, u.w < 0 ,w .u < 0 ?

A: Given data:     The given first vector is: u. The given second vector is: v. The given third vector…

Q: Show that if u and v are orthogonal unit vectors then u x v is also a unit vector.

A: To show: That if u and v are orthogonal unit vector, then prove that u×v is also a unit vector:…

Q: Find projwv. Then decompose v into two vectors, v1 and v2, where v1 is parallel to w and v2 is…

A: If v and w are two non zero vectors, the vector projection of v onto w is  projwv=v.ww2w Substitute…

Q: Prove the stated property of distance between vectors. d(u, w) &lt;= d(u, v) + d(v, w) for all…

A: We have to prove the property of distance between vectors as,

Q: Determine all vectors v that are orthogonal to u.u = (0, −4, 3)

A: Given u=(0,-4, 3)we have to find all vectors v that are orthogonal to u,If u.v =0 then vectors u and…

Q: Let j = | and ü .Write j as the sum of two orthogonal vectors, it -2. %3D an {u} and ¤2 orthogonal…

A: Given:- →y=-69  →u=5-2

Q: Find all vectors in R* that are orthogonal to both 1 1 V1 = and V2 2 -2

A: Linear Algebra: The vectors are: v1=112-2v2=2112Let denote the vector v3=v1v2v3v4

Q: Find a vector that is orthogonal to v =〈1, 2, 1〉 but not to w =〈1, 0, -1〉

A: Given: Find a vector that is orthogonal to v =〈1, 2, 1〉 but not to w =〈1, 0, -1〉

Q: 1. Show that the necessary and sufficient condition for two nonzero vectors A and B to be…

A: As per the company guidelines We can first question only. Please understand. Kindly repost the…

Q: Prove that if the vectors a,b parallel, then āxb=0. are %3|

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Q: When decomposing the vector into the sum of two vectors the first is parallel to the other vector…

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Q: Verify the Pythagorean Theorem for the vectors u and v.u = (4, 1, −5), v = (2, −3, 1)

A: Given   The vector u = (4, 1, −5), v = (2, −3, 1) Pythagorean Theorem for the vectors

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

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Q: Prove that two vectors v = (a, b) and w = (c, d)are perpendicularif and only if ac + bd = 0

A: We prove the above statement using the concept of dot product or scalar product of two vectors. The…

Q: Show that if u and v are n-vectors, then u · v = uT v.

A: Consider the n vectors  u=u1u2u3.un,v=v1v2v3.vn

Q: Show that if the vectors v, w are orthogonal, then ||v + w||^2 = ||v||^2 + ||w||^2.

A: Given: The vectors v, w are orthogonal, then ||v + w||^2 = ||v||^2 + ||w||^2.

Q: Compute the dot product of the vectors u and v, and find the angle between the vectors. u3(-12,0,6)…

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Q: Find a vector v that is perpendicular to the plane containing 3 -() 6 and 0 2

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Q: Determine all vectors v that are orthogonal to u.u = (4, −1, 0)

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

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Q: Prove that the vectors v= and w= are orthogonal.

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Q: Determine all vectors v that are orthogonal to u.u = (11, 2)

A: Now let

Q: Find nonzero vectors u, v, w that are perpendicular to (1, 1, 1, 1) and to each other.

A: Then   -------------(1) Also  --------------------(2)

Q: Realize the Cauchy-Schwarz inequality for the vectors x= ( -1, 2, 4,3), y=(0,-2,1, 4) ∈ R4 and find…

A: I am going to solve the given problem by using some simple algebra to get the required result.

Q: 9. Show that for any real numbers a and b the vectors u = ai and v = bj are orthogonal?

A: Topic:-  vector algebra

Q: Decompose v into two vectors Vị and v2, where vi is parallel to w and v2 is orthogonal to w. v - -2i…

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Q: What can be concluded about non-zero vectors ū, v and w if a) ū ·v = 0 b) ūxi = 0 c) (ū x v) · w= 0

A: As per the company guidelines we are supposed to answer only first 3 subparts of the question. If…

Q: Determine all vectors v that are orthogonal to u.u = (0, 5)

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Q: If u • vx w or v • or w • v equal to zero, then the vectors u, v , and w are coplanar... True False

A: Concept Used: Three vectors are said to coplanar if their scalar triple product is zero That is  [u→…

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: Prove the stated property of distance between vectors.d(u, w) &lt; d(u, v) + d(v, w) for all vectors…

A: Basic logic and properties of vectors can be used to solve the question.

Q: Show that if u + vand ủ – v are orthogonal, then the vectors u and must have the same length

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Q: Find a unit vector that is orthogonal to both u and v.      u = ⟨−8, −6, 4⟩,          v = ⟨10, −12,…

A: Consider the following vectors: u=-8,-6,4v=10,-12,-2 A vector orthogonal to both u and v is the…

Q: Show that if u and v are linearly independent vectors, then so are  u + v and u -v.

A: Given :           u and v are linearly independent vectors.   To prove :               u + v and…

Q: Show that if A⋅(B×C)=0, the three vectors must lie in the same plane.

A: We have to show that if A ⋅ (B × C) = 0 then all three vectors A , B , and C lies in the same plane.…

Q: IH A and B are any two vectors in V, prove that (A- B) x (A + B) = 2(A X B).

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Q: Find a unit vector that is orthogonal to both u = ⟨2, −10, 8⟩ and v = ⟨4, 6, −8⟩.

A: The given vectors are u = ⟨2, −10, 8⟩ and v = ⟨4, 6, −8⟩. Compute the cross product of u and v as…

Q: Find a unit vector that is orthogonal to both u and v. u = (-8, -6, 4) (10, -12, -2) %3D V =

A: Find the unit vector that is orthogonal to both u and v

Q: Find two vectors v and w such that the projection of v onto w is v.

A: Given: There are two vectors v and w such that the projection of v onto w is v. We have to find that…

Q: b) Show that the vector w = u – proj,u is orthogonal to proj, u

A: The expression for the vector projection of u onto v is: projvu=v·uv2v Given: u=3,4,0 and v=-2,3,-6…