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

Find all unit vectors in R^3R3 that make an angle of pi/3π3 with the vectors (1,0,-1)(1,0,1) and (0,1,1)(0,1,1).

1 Answer
Jul 25, 2018

(1/sqrt2,1/sqrt2,0)(12,12,0).

Explanation:

Suppose that, vecv=(l,m,n)v=(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!