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# Rotate a vector to a new position and find its coordinate

Question:

In the Cartesian Coordinate system, point O (0, 0), point A (5, 12), rotate vector OA around point O counterclockwise by 60o to obtain vector OB. What is the coordinate of the point B?

Solution:

The length of the vector OA is

|OA| = Sqrt (52 + 122)
= Sqrt (25 + 144)
= Sqrt 169
= 13

Let the angle a is the angle between the positive x-axis and the vector OA, then

cos a = 5/13
sin a = 12/13

Rotate vector OA around point O counterclockwise by 60o to obtain vector OB. The coordinate of the point B is (x, y). The length of the vector OB is equal to the length of the vector OA. That is, |OB| = |OA| = 13.

Find the x-coordinate of the point B

x = |OB|cos(a + 60o)
= 13 (cos a cos60o - sin a sin60o)
= 13 [(5/13) (1/2) - (12/13) (Sqrt (3)/2)]
= 13 × (1/13) × (1/2) [5 - 12 Sqrt (3)]
= (1/2) [5 - 12 Sqrt (3)]
= - 6 Sqrt (3) + 5/2

Find y-coordinate of the point B.

y = |OB|sin(a + 60o)
= 13 (sin a cos60o + cos a sin60o)
= 13 [(12/13) × (1/2) + (5/13) Sqrt (3)/2]
= 13 × (1/13) × (1/2) [12 + 5 Sqrt (3)]
= (1/2) [12 + 5 Sqrt (3)]
= 6 + (5/2) Sqrt (3)

The coordinate of the point B is (- 6 Sqrt (3) + 5/2, 6 + (5/2) Sqrt (3)). Watch the video for more details.