Find a unit vector that is orthogonal to both i + j and i + k.?

orthogonal-vector

#1

Find a unit vector that is orthogonal to both i + j and i + k.

Answer:

let us assume a vector x = ai+bj+ck is a unit vector as well as orthogonal to i+j and i+k
so then properties of unit vector
a^{2}+b^{2}+c^{2}= 1 eq.1
and orthogonal to i+j and i+k then dot product must be equal to 0
(ai+bj+ck).(i+j) =0 ; a+b =0 : b = -a

(ai+bj+ck).(i+k)=0 ; a+c=0 : c = -a
now putting these values of a and c in eq.1
then a^{2}+(-a)^{2}+(-a)^{2} =1
a=+1/sqrt{3} or-1/sqrt{3}
so the resulting vectors are
x=1/sqrt{3}-1/sqrt{3}-1/sqrt{3}
and y= -1/sqrt{3}+1/sqrt{3}+1/sqrt{3}