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 <em>antiparallel</em> 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