Prove that vector i,j and k are mutually orthogonal using the dot product.

Question

What is actually meant by mutually orthogonal?

Answer 1

mutually orthogonal=the three of them are orthogonal (or perpendicular) to each other,, [angle between them is 90 degrees]

first recall the formula for the dot product. for any given vectors A and B,
A(dot)B=|A||B|cos(theta)
where |A| and |B| are the magnitude of vectors A and B respectively
*note: magnitude means you get the squareroot of the sum of the squares of each element in the given vector.
example, the vector <1,-2,4> has a magnitude of squareroot of (1^2+(-2)^2+4^2) or squareroot of 21

*note: A(dot)B is the sum of the products of the respective elements (x, y and z elements) of two given vectors [this is scalar]
example, A<2,-1,3> and B<-3, 2, 0>
A(dot)B = 2*-3 + -1*2 + 3*0 = -8

going back to your question,
i<1,0,0> ; |i|=1
j<0,1,0> ; |j|=1
k<0,0,1> ; |k|=1
note that i(dot)j is zero, as well as i(dot)k and j(dot)k,,
therefore in the equation, A(dot)B=|A||B|cos(theta)
cos(theta) is equal to zero (since 0/1=0)
therefore, theta you will get is 90 degrees (for i(dot)j, i(dot)k and j(dot)k), which means they are mutually orthogonal to each other,,

so there,, i'm sorry for long explanation..