Consider the following diagram,
The right-hand rule is defined such that it conforms to the right-hand coordinate system. In the right-handed coordinate system with the usual unit vectors (
i,
j,
k) the cross product between
i and
j gives, by definition,
k,
i X
j =
k
Which simply means that if one rotates the x-axis counter-clockwise by \pi/2 such that it lies collinear with the y-axis, then the result is the z-axis. The right-hand rule is commonly used to determine the direction of a cross product, especially in physics problems.
Consider the cross product describe above (
iX
j), which basically means rotating the vector
i (the x-axis) toward the vector
j (y-axis). Now, take your right-hand and keeping your thumb straight (as if giving the 'thumbs up') curl your fingers in the direction which the x-axis is rotating (in this case toward the y-axis). Your thumb should now be pointing straight upwards, in the direction of
k (the z-axis).
Now consider the following cross product,
j X
i
Using the same method as above try to curl your fingers in the direction of rotation (i.e. from the y-axis to the x-axis), you'll probably find that you'll have to turn your hand upside down. If you have done it correctly, your thumb should be pointing direction downwards (towards the negative z-axis). Hence, you have used the right hand rule to determine the cross product,
j X
i = -
k
Indeed, it is very difficult to describe the right-hand rule without demonstrating it. However, I hope you've found my post useful.
