8. We have seen that the matrixR(0) = (cosrotates vectors in R2 counterclockwise by an angle 0. In three dimensions, everyrotation has an axis, i.e. a direction that remains stationary under the rotation.around the y-axissin 0(a) Find the 3 x 3 rotation matrix R₂ (a) that rotates vectors v =- sincosangle a counterclockwise around the z-axiscounterclockwise around the x-axis(i)0Hint: R₂ (a) leaves thez-component of v alone and rotates the x and y components by angle a.(b) Similarly, find the matrix Ry (6) that rotates v by an angle ß counterclockwise(8).-2(:)and the matrix Rz(y) that rotates v by an angle y100by an(c) Verify that R₂ (a), R, (B), and R. (y) are orthogonal matrices.(d) It turns out that any three-dimensional rotation, around any axis, can be writtenas a composition R = R₂(a) R, (B) Rz(y) for some angles a, 8, and y. (We takethis fact for granted.) a, ß, and y are called the yaw, pitch, and roll angles.Explain why the resulting 3 x 3 matrix R is orthogonal. Hint: You do not haveto explicitly calculate the product of these three matrices.

Given: In this problem, the given matrix is R(θ)=cosθ-sinθsinθcosθ , rotates vectors in…

8. We have seen that the matrix R(0) = ( cos sin 0 around the y-axis rotates vectors in R2 counterclockwise by an angle 0. In three dimensions, every rotation has an axis, i.e. a direction that remains stationary under the rotation. sin (a) Find the 3 x 3 rotation matrix R₂ (a) that rotates vectors v = counterclockwise around the x-axis cos angle a counterclockwise around the z-axis (8) 0 Hint: R₂(a) leaves the z-component of v alone and rotates the x and y components by angle a. (b) Similarly, find the matrix R, (B) that rotates v by an angle ß counterclockwise 0 (8). 1 and the matrix R. (y) that rotates v by an angle y 0 1 (6). 0 0 (:) by an (c) Verify that R₂(a), Ry(B), and R. (y) are orthogonal matrices. (d) It turns out that any three-dimensional rotation, around any axis, can be written as a composition R = R₂(a)Ry(B) Rz(y) for some angles a, ß, and y. (We take this fact for granted.) a, ß, and y are called the yaw, pitch, and roll angles. Explain why the resulting 3 x 3 matrix R is orthogonal. Hint: You do not have to explicitly calculate the product of these three matrices.

8. We have seen that the matrix R(0) = ( cos sin 0 around the y-axis rotates vectors in R2 counterclockwise by an angle 0. In three dimensions, every rotation has an axis, i.e. a direction that remains stationary under the rotation. sin (a) Find the 3 x 3 rotation matrix R₂ (a) that rotates vectors v = counterclockwise around the x-axis cos angle a counterclockwise around the z-axis (8) 0 Hint: R₂(a) leaves the z-component of v alone and rotates the x and y components by angle a. (b) Similarly, find the matrix R, (B) that rotates v by an angle ß counterclockwise 0 (8). 1 and the matrix R. (y) that rotates v by an angle y 0 1 (6). 0 0 (:) by an (c) Verify that R₂(a), Ry(B), and R. (y) are orthogonal matrices. (d) It turns out that any three-dimensional rotation, around any axis, can be written as a composition R = R₂(a)Ry(B) Rz(y) for some angles a, ß, and y. (We take this fact for granted.) a, ß, and y are called the yaw, pitch, and roll angles. Explain why the resulting 3 x 3 matrix R is orthogonal. Hint: You do not have to explicitly calculate the product of these three matrices.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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In the equation below, a and c are both vectors of the same size. Given b and d are both scalars, indicated the location of where the dot operator is necessary to perform element-wise operation. Only include dots/periods in the correct locations. A = ((cos (c) + d - a) * (d * a) * tan (b)) / (a^2 + c * exp (a) * sin (b)) Note: Write the equation with the correct position of the dot / periods.
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Let V be the vector space of functions spanned by B = {bị 2 sin r + 6 cos a, b2 = 3 sin a + 7 cos a} where a + %3D C = {ci = sin a, c2 = of coordinates matrix P ne Z is also a basis for V. Find the change 2 = cos a} where a + C+B Ex: 5 P = C-B The coordinates of a function f(x) relative to the basis B are [f(x)]B Find the coordinates of f(æ) relative to Cand find f(æ). Ex: 5 [f(x)]c = f(x) = Ex: 5 sin z + cos a
Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 6t, y = - 4 + 3t O (x, y) = (0, 4) +1 (6,3) ai + bj = -4j + t (6i + 3j) O (x, y) = (0, 4) + t (6, 3) ai + bj = 4j+ t (6i + 3j) O (x, y) = (0, 6) + (-4, 3) ai + bj = − 4j + t (6i + 3j) O(x, y) =(4,0) + t (3,6) ai + bj = −4i + t (3i + 6j) O (x, y) = (6,3)+1(0, - 4) ai + bj (6i + 3j) +t(-4j) (b)x= 4 + 6t,y=4+9t, z = 87t O (x, y, z) = (-4, 4, − 8) +1(-6, ai + bj + ck = 9,7) ( − 4i + 4j - 8 k) + t ( − 6i - 9j + 7k) O (x, y, z) = (4, ai + bj + ck = 4, 8) + t (6, 9, - 7) (4i − 4j + 8 k) + t (6i + 9j – 7k) O (x, y, z) = (6,9, −7) +1(4, - 4,8) ai + bj + ck = (6i + 9j − 7k) + t(4i - 4j + 8k) O (x, y, z) = (-6, 9, 7) + 1 - 4, 4, 8) ai+bj + ck = ( – 6i – 9j + 7k) + t( − 4i + 4j − 8k) O (x, y, z) = (4, 4, 8) + t (6,9,7) ai + bj + ck = (4i +4j + 8 k) + t (6i + 9j + 7k)
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Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 6t, y = - 4 + 3t O (x, y) = (0, - 4) + t (6,3) ai + bj = −4j + t (6i + 3j) O(x, y) = (0, 4) + t (6, 3) ai + bj = 4j + t (6i + 3j) ○ (x, y) = (0, 6) + t ( − 4, 3) ai + bj - 4j + t (6i + 3j) - (x, y) = (-4, 0) + t (3, 6) ai + bj = − 4i + t (3i + 6j) (x, y) = (6, 3) + t(0, − 4) ai + bj = (6i + 3j) + t ( − 4j)
(1) Find the normal vectors of the lines joining the points with the following position vectors (a) 4ị + 7j, 6ị – 3j (b) –2į – 3j, -8į – 7j (c) 2i + j/2, 6+ į/6 (2) Write down in normal form the equation of the line which is (a) parallel to the line r · (5i +3j) = 2 and passes through the point (2, –3); (b) perpendicular to the line 7x + 6y = 1 and passes through the point (1, 1); (c) passes through the points (3, 1) and (–2, –7); (3) Find the equation of the perpendicular bisector of the line join- ing the points with position vectors (a) 3i + 4j, 7i + 1oj (b) –4į – 3j, 2i – 6j. (4) In each of the following cases write down the normal vectors of the given lines and hence find the acute angles between them: (a) 2r – 5y – 3 = 0, 3x+ 7y = 10; (b) 2r + 3y – 2 = 0, 2r – 8y + 19 = 0, (c) 4y – 7y + 13 = 0, 2x – 8y + 19 = 0 (5) Use vector methods to find the position vectors of the points of intersection of the following pairs of lines: (a) r (i – j) +2 = 0, r·(3i+2j)+6 = 0; (b) r· (4i – j) –…