Answer:
- ((3-4√3)/10, (4+3√3)/10)
- ((3+4√3)/10, (4-3√3)/10)
Explanation:
You want two unit vectors that make an angle of 60° with v(6, 8).
Unit vector
After we normalize v to a unit vector, we can find the two desired vectors by adding or subtracting 60° from its angle. The normalized vector v is ...
v = (6, 8)/√(6² +8²) = (3/5, 4/5)
Rotated
This can be rotated 60° CCW by the transformation ...
(x, y) ⇒ (x·cos(60°)-y·sin(60°), x·sin(60°)+y·cos(60°)) = (x-y√3, x√3+y)/2
(3/5, 4/5) ⇒ ((3-4√3)/10, (3√3+4)/10) . . . . 60° CCW from v
It can be rotated 60° CW by the transformation ...
(x, y) ⇒ (x·cos(60°)+y·sin(60°), -x·sin(60°)+y·cos(60°)) = ((x+y√3, -x√3+y)/2
(3/5, 4/5) ⇒ ((3+4√3)/10, (4-3√3)/10)
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Additional comment
When the vector is written as a complex number, it can be rotated by multiplying by 1∠±60° = (cos(60°)±i·sin(60°). That is what the calculator did.
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