198. Apply the 57-degree counterclockwise rotation about the origin to the vectors [1, 0] and [0, 1], then use the image vectors (written as columns) to form the coefficient matrix M for the rotation. Test M by calculating the products M and M Where does this rotation send the vector [3, 1]? Does M, when applied to do its job correctly? 1

Q: 2- We can write the vector x = (19,–1,–) in R³ as a linear %3D combination of the vectors x1 = (4,2,…

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Q: Compute the new position of the square [A(19,19), B(29,19), C(29,29), D(19,29)] after its rotation…

A: ABCD is a square , we need to rotate it clockwise along with centroid.

Q: Write x as the sum of two vectors, one in Span fu1,u2,uz} and one in Span fu4}. Assume that fu, .u4}…

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Q: 31. The vector perpendici:!at to both 3i –-6j – 4k and -3i + 2j + 2k is A A -4i + 6j + 12k B - 4i +…

A: Cross product

Q: Find the coordinate vector [x]gof x relative to the given basis B= {b1,., bn}. -[-]»= [-{]==(:)

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Q: Consider the vectors u = (-2, 1, –3) and i = (1, –2, – 1). Find the angle between u and ū. (b) Find…

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Q: Find the vector projection of b = 6î + 3ĵ + 2k onto å = î – 2ĵ – 2k.

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Q: .

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Q: P, find the change-of-coordinates matrix from the basis In B = {1 – 2t + t², 3 – 5t + 41², 2t + 3t²}…

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Q: (i, 2, 3i), v = Co over C. Are these vectors linearly independent? Can they form a basis 5.…

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Q: 6. Consider the vectors u = (1,–2), v = (4,3), and w = (9,4). Find scalars k and k2 such that w =…

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Q: What is the matrix representing a translation by vector [−1, 2, −1]? What is the matrix representing…

A: The given matrix is -12-1. The rotation angle at 90°. So if the rotation is θ around Y axis is,…

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A: Given p=5x2+2x+3 and basis  p1=1p2=1+xp3=1+x+x2

Q: Find out Vector S= { V, V2, V3} is the Basis por R. if VI : ( 3,1,-4), Vz C2,s,6), V3 - C1,4,8) %3D…

A: Here we have to find whether v1 , v2 , v3 are basis of R3 or  not.

Q: 3- The scalar component of the vector F in the direction of K is (K.F А- .K, В- K C-

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Q: (b) Prove that under the assumption of very small rotations and right-hand convention, the 3-D…

A: To show: The 3D rotation matrix about the three fixed X-, Y-, and Z-axes, under the assumption of…

Q: Find the image of the vector (1, 1, 1) for the rotation.120° about the x-axis

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Q: Let S = and T = be ordered bases for R2. Let v = and w = (a) Find the coordinate vectors of v and w…

A: Given: S=12,01 T=11,23 v=15 w=54 (a) Let v=a11+b2315=a11+b23. Then, a+2b=1 and a+3b=5, On…

Q: O Explatn why each of the followng statements True or False. a) ar[? ó] represents rotation…

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Q: Suppose T(v1) = w1 + w2 + W3 and T(v2) = w2 + w3 and T(v3) = W3. Find the matrix A for Tusing these…

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Q: Q1. If O (0,0,0) in (x,y,z) 3-D space and r(2,5,1) draw the vector in 3D and find its direction.

A: Plot point in 3D

Q: Write x as the sum of two vectors, one in Span fu1,u2,u3 and one in Span fu4. Assume that fu1,...4is…

A: Let x = v + wWhere v = a1u1 + a2u2 + a3u4        and w = a4u4.

Q: Suppose {u, } form an orthogonal basis in R2 and y = 2ū - 7 where u = (8) 6 If W = Span{}, what is…

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Q: 1. Let S2 {B1, B2} be an ordered basis set of vectors in 2-D. (a) Let ß1 = i, B2 = j in Cartesian…

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Q: Find the coordinate vector of [4, -2, 1] relative to the ordered basis B= ([0, 1, 1], [2, 0, 0], [0,…

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Q: Find the coordinate vector of p(x) in P., relative to the basis S= {p,.P.-P;}, where p(x) =…

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Q: 13. In P2, find the change-of-coordinates matrix from the basis B = {1 – 21 + t², 3 – 5t + 41², 2t +…

A: To find - In ℙ2, find the change-of-coordinates matrix from the basis B = 1 - 2t + t2, 3 - 5t + 4t2,…

Q: orthonormal 3D vectors

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Q: 3 Find the coordinates of y 7 relative to the orthogonal basis Uj = U2 = U3 = -2 -4 10 2 2 4 252

A: The solution is given as

Q: 4 Suppose v1, V2, V3 are vectors in R*, and the matrix A = v, v2 V3 has rank 1 W = Span{v1, V2, V3}…

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Q: Find the matrix that produces the rotation. Then find the image of the vector (1, −1, 1).45° about…

A: Given  The vector (1, −1, 1) and 45° about the z-axis is

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Q: Write the Rodrigues vector and the quaternion corresponding to the rotation matrix - cos(T/3)…

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Q: (a) Specify the Euler angle rotation matrices Rx, Ry and Rz for rotation about the X, Y and Z-axis…

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Q: Find a vector orthogonal to (0, 1, 7) and (2i – j+ 4k) 11i + 14j - 2k 2i - j+ 11k -j + 28k 4i + 14j…

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Q: Let T be the flection about the line -2x + ly = 0 and T, be the reflection about the line 2x + ly =…

A: The matrix of reflection about a line y=mx is. A=11+m21-m22m2mm2-1 Given that T1 is a reflection…

Q: Find the scalar and vector projections of d = (4, 2, –5) on b = (3, –1, 6).

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Q: 1 -1] 5 -4 2 1 -3 1 -2 -1 3 2 16

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Q: Determine the matrix that produces the pair of rotations. Then find the image of the vector (1, 1,…

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Q: Find the image of the vector (1, 1, 1) for the rotation.60° about the x-axis

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Q: If u = (3, −1, 4) and v = (−2, 3, 5), the scalar projection of y onto v is:

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Q: Does the vectors v1 = (-3, 7) and v2 = (5, 5) form a basis for R2?

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Q: Use matrices to transform the vector .3 by a reflection across the x-axis, a horizontal compression…

A: Given:     Vector : ⟨ 7 , 3 ⟩     Reflection across x - axis   Horizontal compression = 1/2…

Q: 1. Find the vector x determined by the given coordinate vector [x]g and the given basis В. 8 B =…

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Q: Find the orthogonal projection of the vector b =-1 on Nul(A), where -2 -2 3 -10 A = 1 -1 2 3 1 -3 3…

A: For the solution follow the next steps.

Q: Find the coordinate vector for the vector v = (3,4,3) relative to the basis S={(3,2,1), (-2,1,0),…

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