How to determine orthogonal vector in 2d space?

Question

Given a vector V1 (a, b) in 2d space , what is the simplest way to get unit vector V2 that is orthogonal to V1?

Answer 1

The dot product of orthogonal vectors is zero. In two dimensions the slopes of perpendicular lines are negative reciprocals. Switch the x and y coefficients and change the sign of one of them.

For vector V1 = , the reciprocal V2 = . Now make it a unit vector by dividing by the magnitude.

V2 = / √(a² + b²)

As a check take the dot product.

V1 • V2 = • / √(a² + b²)

V1 • V2 = | ab - ab | / √(a² + b²) = 0 / √(a² + b²) = 0

So they are orthogonal.

Answer 2

You get orthogonal vector when scalar product = 0
for V1(a,b) and V2(a',b') :
aa'+bb'=0

Once you have orthogonal vector, unit vector is given by :
a'²+b'² = 1

Example :
given vector v1(1,1)
if a'=1 then b' = -aa'/b = -1

a'²+b'² = 2
So you have to take (sqrt=square root)
a' = 1/sqrt(2) and b'=-1/sqrt(2) in order to have a'²+b'²=1