Answer:
39.81°
Step-by-step explanation:
There are numerous ways to find the angle between two vectors. Perhaps one of the easiest is to subtract the angle of one from the angle of the other.
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angle difference
∠v = arctan(6/-5) = 180° -50.1944° = 129.8056°
∠w = arctan(8/0) = 90°
Then the angle difference is ...
∠v -∠w = 129.8056° -90° ≈ 39.81°
The angle between the vectors is about 39.81°.
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angle of ratio
The angle can also be found by finding the angle of the ratio of the vectors. This is most easily done by representing the vectors as complex numbers.
v/w = (-5 +6i)/(8i) = (-5 +6i)(-8i)/(64) = (48 +40i)/64 = (6+5i)/8
∠(v/w) = arctan(5/6) = 39.8056° ≈ 39.81°
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Additional comment
All spreadsheets and many calculators provide an ATAN2(x, y) function that gives the arctangent of y/x in the correct quadrant. This form of the arctangent function is able to deal properly with the case of x=0. In the above, we show arctan(8/0) = 90°. This is effectively atan2(0, 8) = 90°.
