P( 1, 1) is in a circle.

Vector PA. Vector PB=-|PA|*|PB|

Using the chord intersection theorem of a circle

It is known that the center of the circle is O and the radius is 2.

Connect the OP, and the intersection of the straight line OP and the circle is set to c and d.

Then |OP|=√2,

∴ |PC|=2-√2，|PD|=2+√2

∴ |PA|*|PB|=|PC|*|PD|=4-2=2

Vector PA. The vector PB=-|PA|*|PB|=-2.