If a point (x, y) in the plane is rotated counterclockwise about the origin through an angle of 60°, its new coordinates (x', y') are given by = S *:: and a =an o = / - 0.0660. where S is the 2 × 2 matrix a -b 3 z 0.8660. 2 (a) If the point (6, 8) is rotated counterclockwise through an angle of 60°, what are its (approximate) new coordinates? (Round your answers to four decimal places.) (x, y) (b) Multiplication by what matrix would result in a counterclockwise rotation of 105°? (Express the matrices in terms of S a and the matrix R which is a 2 x 2 matrix a -a where a = / 0.7071.) [HINT: Think of a rotation through а а 105° as a rotation through 60° followed by a rotation through 45°.] (c) Multiplication by what matrix would result in a clockwise rotation of 60°? (Express the matrix in terms of S.)

If a point (x, y) in the plane is rotated counterclockwise about the origin through an angle of 60°, its new coordinates (x', y') are given by = S *:: and a =an o = / - 0.0660. where S is the 2 × 2 matrix a -b 3 z 0.8660. 2 (a) If the point (6, 8) is rotated counterclockwise through an angle of 60°, what are its (approximate) new coordinates? (Round your answers to four decimal places.) (x, y) (b) Multiplication by what matrix would result in a counterclockwise rotation of 105°? (Express the matrices in terms of S a and the matrix R which is a 2 x 2 matrix a -a where a = / 0.7071.) [HINT: Think of a rotation through а а 105° as a rotation through 60° followed by a rotation through 45°.] (c) Multiplication by what matrix would result in a clockwise rotation of 60°? (Express the matrix in terms of S.)

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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Question

If a point (x, y) in the plane is rotated counterclockwise about the origin through an angle of 60°, its new coordinates (x', y')
are given by
a -b
where S is the 2 × 2 matrix
b
1
and b =
2
and a =
* 0.8660.
a
(a) If the point (6, 8) is rotated counterclockwise through an angle of 60°, what are its (approximate) new coordinates?
(Round your answers to four decimal places.)
(x, y) =
(b) Multiplication by what matrix would result in a counterclockwise rotation of 105°? (Express the matrices in terms of S
a -a
and the matrix R which is a 2 x 2 matrix
where a =
x 0.7071.) [HINT: Think of a rotation through
a
105° as a rotation through 60° followed by a rotation through 45°.]
(c) Multiplication by what matrix would result in a clockwise rotation of 60°? (Express the matrix in terms of S.)

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