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.
12 thoughts on “cross or vector product of unit vectors”
What is the answer of
i . j . k = ?
i x j x k = ?
Please tell me the result of -jxk
dot product of adjacent vectors is zero. i.e i.j=0 then i.j.k=0…
cross product of adjacent vectors gives the next one ixj=k then ixjxk=kxk=0.
(i.j).k=0 as 0*k=0
Vector multication and dot product
I understand i x i = 0 and i · i = 0,
but how does i² = -1 ?
Clearly, it’s neither i x i nor i · i.
Then what is it?
Thanks for comment. Where is i square mentioned?
There, i is an imaginary number which is the root of -1. So, i*i=-1.
well the i that u are talking about is imaginary no which is root (-1) here i is aunit vector we are talking about.just same symbols used for different things
ixi=0 ; here our ” i ” is a unit vector.
i² = -1 ; here our ” i ” is an imaginary number.