# cross or vector product of unit vectors

Last time I have written about the dot product of unit vectors that is:

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.

What is the answer of

i . j . k = ?

And

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

ixjxk=0

0

(i.j).k=0 as 0*k=0

ixjxk=kxk=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.

You are talking about complex, not vector, i. e. i = sqrt(1)

First of all when i is an imaginary number or complex number or unreal number because i=√-1

And that’s why it is when i.i=i^2=-1

You are talking about complex, not vector, i. e. i = sqrt(-1).

As such isqr = -1