The position vector \overrightarrow{O P} of point {P}(20,10) is rotated anti-clockwise in {X}- {Y} plane by an angle \theta=30^{\circ} such that point {P} occupies position {Q}, as shown in the figure. The coordinates (x, y) of Q are

Correct Answer is : (12.32,18.66)

GATE Mechanical Engineering (ME) 2019 Shift 1 Solved Paper

Question : 34 of 65

Marks: +1, -0

The position vector

{O P}↖{→}

of point

{P}(20,10)

is rotated anti-clockwise in

{X}- {Y}

plane by an angle

θ=30^{∘}

such that point

{P}

occupies position

{Q}

, as shown in the figure. The coordinates

(x, y)

Q

are

Solution:

Rotational matrix for a point in 2D coordinated by 𝜃 in anti-clockwise rotation

R=[\table \cos θ , -\sin θ ; \sin θ , \cos θ]

Point

_{Q}=R\{\table 20 ; 10\}=[\table \cos θ , -\sin θ ; \sin θ , \cos θ]\{\table 20 ; 10\}

Point

_{Q}=[\table \cos 30 , -\sin 30 ; \sin 30 , \cos 30]\{\table 20 ; 10\}

Point

_{Q}=[\table 0.866 , -0.5 ; 0.5 , 0.866]\{\table 20 ; 10\}=\{\table 12.32 ; 18.66\}

Point

_{Q}=(12.32,18.66)

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