i.i=1

j.j=1

k.k=1

i.j=j.k=k.i=0

Today I will discuss about the cross product of unit vectors:

i x j = k

j x k=i

k x i =j

j x i= -k

k x j= -i

i x k= -j

and

i x i = j xj = k x k =0

Do you know how the above results come? If your answer is no, then let us discuss it:

I have already explained in my earlier articles that cross product or vector product between two vectors A and B is given as:

A.B = AB sin θ

where θ is the angle between A and B. A and B are magnitudes of A and B.

As i the unit vector along x axis

Therefore i x i = 1sin 0

This is because, first i is the unit vector of A along x axis and second i is the unit vector of B along x axis.

Therefore two unit vectors must be in the same direction that is x direction so the angle between them will be 0 degree. As i and i are unit vectors therefore there magnitudes will be unity. As sin 0 is 0,

Therefore above equation will become: i x i =o

Similarly

j x j =0

k x k =0

Then why i x j =k,

This is because, i along x axis and y along y axis, thus, angle between them will be 90 degree. As sin 90 = 1. As curl or rotation of two vectors give the direction of third vector

Therefore, i x j = 1 sin 90 k

i x j = k

but j x i = – k because now the direction is reversed or due to vector identity A x B is not equal to B x A.

Similarly

j x k=i and k x j = -i

k x i=j and i x k = -j

Note: I hope that now you can understand and explain everything about the cross or vector product of two unit vectors.

Reference: These articles are referred from my authored book “concepts of electromagnetic field theory” having ISBN 978-81-272-5245-8. In case of any doubt in this article or any other EMFT or physics related article, kindly post in the comment section.

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