Consider two vectors →A=Ax^i+Ay^j+Az^k and →B=Bx^i+By^j+Bz^k. Scalar product of these two vectors can be found out by following two different formulae. →A⋅→B=AxBx+AyBy+AzBz. →A⋅→B=ABcosϕ Prove that the above two formulae for scalar product of vectors are equivalent.

Q: (1, Find a vector orthogonal to both (3, 2, 0) and to (0, 2, -1) of the form

A: The given question is related with vector space. We have to find a vector orthogonal to both 3 , 2 ,…

Q: Find a vector of norm 1 that is orthogonal to the three vectors ü = (2,1,-4,0), v=(-1,-1,2,2),…

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Q: What does it mean for one vector to be a scalar multiple of another vector? How can you tell whether…

A: What does it mean for one vector to be a scalar multiple of another vector? How can you tell whether…

Q: Find linearly independent vectors which generate al lutions of Ax = 0.

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Q: Find two unit vectors orthogonal to both (2, 6, 1) and (-1, 1, 0). (smaller i-value) (larger…

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Q: If a vector is multiplied by a positive scalar, how is the result re-lated to the original vector?…

A: Explanation:

Q: How do I show that the vectors (1, 2), (0, 1), (-3, 4) are not linearly independent?

A: To verify that the three given vectors in the plane are linearly dependent

Q: cannot be found in such a general cas explain why. In either case, justify you 2) Let a and b be two…

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Q: Find two unit vectors orthogonal to both (2, 7, 1) and (-1, 1, 0). (smaller i-value) (larger…

A: Given that the vectors are 2,7,1 and -1,1,0.It is required to find the two unit vectors orthogonal…

Q: What is the scalar projection of á = (1,0, 1) onto b = (0, 3, 4)?

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Q: Prove that u · v = 1/4 llu + vll2 -1/4llu - vll2 for all 4 4 vectors u and v in Rn.

A: Given Data The equation to prove is u∙v=1/4||u+v||²-1/4||u-v||².   Evaluate the right hand side for…

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

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Q: The scalar projection of vector A onto another vector B will be the negative of that scalar…

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Q: Find two unit vectors orthogonal to both u = (1, 2, – 3) and = (0, 1, 2). %3D unit vectors =

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Q: 2- Are the vectores [4,2,6] and [6, 3, 9] collinear? How do you know?

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Q: 3. Give two nonparallel vectors that are both orthogonal to (0, 5, 0).

A: Given A = (0,5,0) Let other two vectors be of type B = (a,b,c) A.B = a.0+b.5+c.0 A.B = 5b For…

Q: Find a vector of length 5 that is orthogonal to both vectors ⟨4,−1,0⟩ and ⟨−3,2,−1⟩

A: The vector orthogonal to two vectors u,v is = ± (u×v)/ |u| |v| Here u =  ⟨4,−1,0⟩ , v =…

Q: Find a vector of length 2 orthogonal to the plane 3x – 4z = 5. %3D

A: Here we have to find the vector of length =2 and  orthogonal to the plane 3x - 4z = 5 Let the vector…

Q: Find two vectors of norm 1, orthogonal to all the vectors u = (2, 1, 4, 0), v = (−1, −1, −2, 2), w…

A: Given data:- u=2,1,4,0v=-1,-1,-2,2w=3,2,5,4

Q: It is possible for two nonzero vectors to have a dot product of 0. True False

A: Let u and v be two nonzero vectors

Q: Project the first vector orthogonally into the line spanned by the second vector. 2 3 -2

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Q: If the cross product of two nonzero vectors is the zero vector, then each of the two vectors is a…

A: Explanation of the answer is as follows

Q: the component of vector a are ax=2.0 units and ay=3.0units while that of another vector b are Bx=…

A: Given components of two vectors: Components of a are: ax=2.0 units and ay=3.0 units, Components of b…

Q: The terminal point of the vector v = 3i +5j – 4k with initial point (2, –1, 4) is

A: Follow next step

Q: Find a vector orthogonal to both (-3, 2,0) and to (0, 2, 1) of the form (1,

A:

Q: 2- Concisely describe what the following terms mean for a set of vectors and how we quickly check…

A: Define: 1. Collinear vectors 2. Coplanar vectors 3. Linearly independent vectors 4. Linearly…

Q: Find two unit vectors orthogonal to both (3,2,1) and (–1,1,0).

A: The vectors 3,2,1 and -1,1,0

Q: Obtain a set of orthonormal vectors by Schmidt method: u = (1, 1,0,1), uz = (2,0,0, 1), uz (0, 2,3,…

A: According to gram Schmidt method  orthogonal vectors can be obtained as: x1=u1x2=u2-<u2,…

Q: Find two vectors that are orthogonal to ⟨0, 1, 1⟩ and to each other.

A: Vector: Let the two vectors are v and w. We have to find the two vectors which are orthogonal to…

Q: Find two unit vectors orthogonal to both (5, 8, 1) and (-1, 1, 0). (smaller i-value) (larger…

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Q: The terminal point of the vector v = 3i -j+ 5k with initial point (2, 5, -5) is Do not leave any…

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Q: Compute for the cross product of the following pairs of vectors (-4, 3, 0) and (6, 2, -5). Then,…

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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: Exercise 4. Is it possible to have nonzero vectors u, v in V with uv = 0? Why or why not? (Here the…

A: Given: 4) V is a subset of vectors in Hamilton's quaternions. To explain: Why or why not there are…

Q: Find the scalar and vector projections of b onto a. a = 3i - j + 6k,     b = j + (1/2)k compab =…

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Q: Find the scalar and vector projections of b onto a. а - (-4, -3), ь - (6, 0) compab projąb

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Q: Find two unit vectors orthogonal to both (6, 9, 1) and (-1, 1, 0). (vector with the smaller…

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Q: 8. Show that the cross product u x v of the vectors u = (-3, 5, -4) and v = (-3, -5, -5) is…

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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: Prove that llu + vll2 + llu - vll2 = 2llull2 + 2llvll2 for all vectors u and v in Rn

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Q: Find two unit vectors orthogonal to both (2, 3, 1) and (-1, 1, 0).

A: Unit Vectors orthogonal to vectors  <2,3,1> and <-1,1,0>…

Q: Calculate Using 1.Gm a) Vector sduion b) scalar solution c) Ic soluñ0 3.Sm

A: ..

Q: If a vector is multiplied by a positive scalar, how is the result related to the original vector?…

A: Solution:- If a vector is multiplied by a positive scalar, how is the result related to the original…

Q: (i) Compute each Ax by dot products of the rows with the column vector: (a) 2 1 1 2 3 [10 2 1 1 2 3…

A: Since, you have asked two questions, we are attempting only the first question for now. If you want…

Q: 3. Show that that the vector u = -5 is a linear combination of the vectors vị = 3 U2 = and v3 %3D

A: Given vectors are u=   2-5  3, v1=  1-3  2, v2=  2-4-1 and v3=  1-5  7   To show that u is a linear…

Q: Let u, = (1,1,1)", u2 = (1,2,2)T and uz = (2,3,4)", find the coordinates of the vectors (2,3,2)" and…

A: Solving by linear combination method

Q: Find a vector of norm 1 that is orthogonal to the three vectors u = (2,1,-4,0), =(-1,-1,2,2),…

A: The solution is given as

Q: Find two orthogonal vectors in the plane x + y + 2z = 0. Make them orthonormal.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Express the product Ax as a linear combination of the column vectors of A. 1 21 3 + 3 +3 2.

A: The given product, A2×2x2×1=12-3432 Which will result into a vector of order 2×1