5. The vector A=5i+6j is rotated through an angle 45degree the z-axis in the anticlockwise direction. The resultant vector is
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Explanation:
The vector and rotation are both in the x-y plane. So a rotation about the z-axis is the same as a rotation about the origin in the x-y plane.
The matrix for rotation through an angle θ in the plane is
[cos θ -sin θ]
[sin θ cos θ ]
So the resultant vector is
[cos 45 -sin 45][5]
[sin 45 cos 45 ][6]
= (1/sqrt 2)[1 -1][5]
………………. [1 ….1][6]
= (1/sqrt 2)[-1]
…………………[11]
= (-1/sqrt 2, 11/sqrt 2)
= approx. (- 0.7071, 7.7782)
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