Rotating a point around another using vectors. - method is wrong?

Question

The question: On what coordinates will point B = (7,5) land if it is rotated around point A = (0,0) by 110 degrees?

My workings out gave me the answer B will land on point (-7.09, 4.87). I got this answer by figuring out that the vector from A to B is (7 , 5) and therefore the direction angle of the vector AB is tan^-1(5/7) = 35.54° by using pythagoras on vector AB. Thus the direction angle of AB1, where B1 is the new point on which B will land, is 110 + 35.54 = 145.54 degrees. Then, one can use the rule that the unit vector in the unit circle is (cos(v), sin(v)), and by finding out the length of the vector AB (√(7^2+5^2 = √74)), you can apply this rule to the new AB1 vector and you get that the vector AB1 = √74 * (cos(145.54°) , sin(145.54°)). This, in turn, is equal to AB1 = (-7.09, 4.87). Now that we have the vector of AB1, we can use this to find point B1, by adding the values inside vector AB1 to the point we are rotating B around (A = (0,0)). Therefore B1 = (-7.09, 4.87). Given that this is such a complex method, I have doubts about the validity of my answer. Is this right or have I gone wrong somewhere?

Answer 1

(7,5) rotated 110 degrees about the origin

(7,5) in polar coordinates = (sqr(49+25), tan^-1(5/7)

= about (8.6, 35.54)

rotate 110 is (8.6, 35.54+110)

= (8.6, 145.54)

convert back to rectangular coordinates

(8.6cos145.54, 8.6sin145.54)

=(-7.09, 4.87)

it may help to plot the points and look at a graph

if you rotated clockwise the new point

is

about (8.6, 35.54 -110)

= (8.6, -74.46)

convert to rectangular coordinates to get

= (8.6cos-74.46, 8.6sin-74.46)

= (2.23, -8.29)

rounded off to 2 decimal places

Answer 2