What is a "negating vector"?  This is a vector that cancels out the effects of some other vector(s).
For example, if you are in a tug of war against a friend, let's say your friend pulls with 100 Units of force on the rope.  To stop the rope from moving, you pull your end with 100 Units as well.

Your two vecors cause the system (you, friend, and rope) to not move.  This situation is called static equilibrium.
The situation looks like this... Each arrow is the same length.

WITH a little imagination, you can extend this situation....
If three people with equal strength pull on a tent stake, one in each of these directions: north, east, and west.... Where would a fourth person have to pull (with the same strength) in order to create static equilibrium?
Hopefully, your common sense voice is saying, "SOUTH!!!".
 
 

Here is another easy example....
Vector A is 40 units long and points at +45°. Find length and direction of a new vector "B" that would cancel out vector A.

Mathematically speaking, we want this statement to be true: A=-B

COMMON SENSE METHOD: Vector B must be same strength as A, so is 40 units, and must point in opposite direction as Vector A, so is at 45° + 180° = 225°

MATH METHOD: find Ax and Ay, then use their opposites to get "B"
Ax = 40cos45 = 28.28 units ===> therefore B has Bx = -28.28 units
Ay = 40sin45 = 28.28 units ===> therefore B has By = -28.28 units

Use Pythagorean Theorem to get length of B.... sqrt(-28.28² + -28.28²) = 40 units
Use tangent to get angle... q = inv tan(-28.28/-28.28) = 45°+180° = 225°
 

This technique can be applied to numerous situations, many we will see later when we discuss momentum and force. For now, log into Lecture OnLine and solve: "A+B=-C" and "vector T/F"