cross or vector product of unit vectors
Last time I have written about the dot product of unit vectors that is:
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
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
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.
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.