Question

Let v = (v1, v2) be a vector in R2. Show that (v2, −v1) is orthogonal to v, and use this fact to find two unit vectors orthogonal to the given vector.
v = (4, 3)

Answer 1

The two given vectors are orthogonal because

$(v_1,v_2)\cdot(v_2,-v_1)=v_1v_2-v_2v_1=0$

This means that the vector (3, -4) is orthogonal to (4, 3); normalize it to get the unit vector,

(3, -4) / sqrt(3^2 + (-4)^2) = (3/5, -4/5)

You can get another unit vector orthogonal to (4, 3) by multiplying this unit vector by -1, (-3/5, 4/5), which points in the direction opposite from (3/5, -4/5).