What are the unit vectors that make a certain angle with two other vectors?

Question

Find all unit vectors in #R^3# that make an angle of #pi/3# with the vectors #(1,0,-1)# and #(0,1,1)#.

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Answer 1 · 2018-07-25T18:29:51

# (1/sqrt2,1/sqrt2,0)#.

Suppose that, #vecv=(l,m,n)# is the reqd. vector.

Since #||vecv||=1, :., l^2+m^2+n^2=1...................(star_1)#.

If #vecu=(1,0,-1) and vecw=(0,1,1)#, then, by what is given,

#/_(vecv,vecu)=pi/3#.

#:. vecv*vecu=||vecv||*||vecu||*cos(pi/3)#.

#:. (l,m,n)*(1,0,-1)=1.sqrt(1+0+1)*1/2#.

#:. l-n=1/sqrt2.................................(star_2)#.

Similarly, from the given #/_(vecv,vecw)=pi/3#, we get,

# m+n=1/sqrt2................................(star_3)#.

Utilising #(star_2) and (star_3)" in "(star_1)#, we get,

# (n+1/sqrt2)^2+(1/sqrt2-n)^2+n^2=1#.

#:. 3n^2=0, or, n=0#.

#:. vecv=(l,m,n)=(1/sqrt2,1/sqrt2,0)#, is the desired vector!