Urgent math question perpendicular unit vectors 2D plane

Find the unit vector perpendicular to the vector (4i - 3j).
ANSWER -1/5 (3i+4j) or -1/5 (3i+4j)

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please explain fully how i go about doing this question
and why are there two cases to the answer

i can find the unit vector if its parallel but i have no clue when its perpendicular (school me please)
thank you :)

Let ai + bj be a unit vector perpendicular to 4i - 3j.
Since the vectors are perpendicular the scalar product is 0.
(ai + bj) . (4i - 3j) = 0
4a - 3b = 0
Also, since ai + bj is a unit vector:
a² + b² = 1
Substitute b = 4a/3 into a² + b² = 1.
a² + 16a²/9 = 1
25a² = 9
a = 3/5 or -3/5
Substitute a = 3/5 and a = -3/5 into 4a - 3b = 0
12/5 - 3b = 0 or -12/5 - 3b = 0
b = 4/5 or b = -4/5
So there are two possible answers: (3/5)i + (4/5)j and the inverse of that.

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The answer should be 1/5 (3i+4j) or -1/5 (3i+4j).

1) Draw the given vector.
2) Find the angle (theta) between vector and x-axis.
3) Find the magnitude (length) of vector.
4) Draw a line through origin and perpendicular to original vector.
5) The angle that's between x-axis and perpendicular line is 90 - theta = phi.
6) Use either phi or theta to calculate the vector w, which has magnitude of 5.
7) The other vector is the same vector, but negative.

A unit vector is one whose length ( modulus) is 1. So if you have a vector v, the corresponding unit
vector is (1/|v|)v. The two answers you give are not different but maybe you mean one is +
and the other -. In that case then the answer is correct . The unit vector is unique apart from sign.
Here, v=4i-3j, so |v|=sqrt(4^2+(-3)^2)=5 and v/|v|=(1/5)v=(1/5)(4i-3j) but you can also have (-1/5)(4i-3j).
Not that v and -v have the same length but are in opposite directions.