Vectors - Dot Product

[mykonos]

Given the points O(0,0) and P(5,5) describe the set of points Q such that
a) vector OP•vector OQ = 0
b) vector OP•vector OQ = -?2/2
c) vector OP•vector OQ = -?3/2

Spoiler: :5kalz5ud

Answers:
a) y= -x
b) -x-?2/10
c) -x-?3/10[/spoiler:5kalz5ud]

I dont understand how to arrive at these answers

 

[daon]

A zero dot-product signifies orthogonality. Hence, rotation by pi/2 radians gives you an orthogonal vector. You may rotate 4 times before arriving back at your original vector obtaining two vectors orthogonal (pi/2 radians and 3pi/2 radians), but you have infinitely many vectors at each angle which satisfy this (think vectors of different lengths).

<(5,5), (q1,q2)>=0 if and only if 5*q1+5*q2 = 0 => (q1 = -q2) or (q2 = -q1)

I broke it into two cases to make the point that the first would be rotation by pi/2 radians and the second by 3pi/2. They are equivalent however (i.e. you only need one of them) if you consider all real numbers instead of just positive ones.

Thus all points that lie on the line defined by real numbers satisfying q1=-q2.

The second is done similarly...

<(5,5), (q1,q2)> = -sqrt(2)/(2) ==> 5*q1+5*q2 = -sqrt(2)/2 ==> q1+q2 = -sqrt(2)/(5*2) => q1 = -q2 + -sqrt(2)/10

As above, it is a line.