Any vector cross any other vector implies, both the vectors are having there tail point fixed at a same point and one among the two is rotating either near or away from other vector, such that the resultant of this setup is another vector in direction perpendicular to both the given vectors.
If the mathematical statement says, a x b = c , then it means that vector a is at fixed point and vector b is moving towards or away from it in clockwise sense, such that the resultant c is mutually perpendicular to both a & b , and is upward or along positive direction i.e. along positive z-axis.
Whereas the statement, a x b = -c , means the same setup is rotating in opposite sense i.e. anti-clockwise sense and the resultant c is downward.
Now let us consider the required speacial case i.e. a x b = 0 :
In such case, our resultant i.e. c is 0 , which means that our resultant neither lies on positive z-axis nor negative z-axis. Also it implies that there is no rotation between the two vectors a & b .
No rotation is possible only when the two vectors i.e. a and b are either at the same position (i.e. coinciding) or are parallel. Implies the direction of both the vectors are same. Which says that a = b (direction only).
Now after calculation of the magnitude of each vector i.e. a and
b , if the magnitude of both the vectors comes out to be same, then it proves that both the vectors are also equal in magnitude.
Hence now both the vectors can be said “Equal” i.e. vector a = vector b.
Answers & Comments
Any vector cross any other vector implies, both the vectors are having there tail point fixed at a same point and one among the two is rotating either near or away from other vector, such that the resultant of this setup is another vector in direction perpendicular to both the given vectors.
If the mathematical statement says, a x b = c , then it means that vector a is at fixed point and vector b is moving towards or away from it in clockwise sense, such that the resultant c is mutually perpendicular to both a & b , and is upward or along positive direction i.e. along positive z-axis.
Whereas the statement, a x b = -c , means the same setup is rotating in opposite sense i.e. anti-clockwise sense and the resultant c is downward.
Now let us consider the required speacial case i.e. a x b = 0 :
In such case, our resultant i.e. c is 0 , which means that our resultant neither lies on positive z-axis nor negative z-axis. Also it implies that there is no rotation between the two vectors a & b .
No rotation is possible only when the two vectors i.e. a and b are either at the same position (i.e. coinciding) or are parallel. Implies the direction of both the vectors are same. Which says that a = b (direction only).
Now after calculation of the magnitude of each vector i.e. a and
b , if the magnitude of both the vectors comes out to be same, then it proves that both the vectors are also equal in magnitude.
Hence now both the vectors can be said “Equal” i.e. vector a = vector b.
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