if C is the vector sum of A and B C=A+B what must be true about directions and magnitude of A and B if C=A+B? what must be true about the directions and magnitude of A and B if C=0​

a) Magnitudes: , , ; Directions: for . Undefined for , for . Undefined for , for . Undefined for .

b) Magnitudes: , , ; Directions: , is undefined.

a) Let suppose that , and , where is known as Vector Zero. By definitions of Dot Product and Inverse Trigonometric Functions we derive expression for the magnitude and directions of , and :

Magnitude ()

Magnitude ()

Magnitude ()

Direction ()

for . Undefined for .

Direction ()

for . Undefined for .

Direction ()

for . Undefined for .

Please notice that is the Vector Unit.

b) Let suppose that and and . Hence, . In other words, we find that both vectors are antiparallel to each other, that is, that angle between and is 180°. From a) we understand that , , but .

Then, we have the following conclusions:

Magnitude ()

Magnitude ()

Magnitude ()

Directions (, ):

Direction ():

Undefined