Finding Unit Vectors for Angle of pi/3 in R3

[gunnar]

There are two vectors (1,0,-1) and (0,1,1)
I need to find all the unit vectors x in R3 that make an angle of pi/3 with each of the vectors above.

Can someone please help with this problem?

 

[vincentchan]

hint:
use dot product to find the angle between vectors

 

[Hurkyl]

You should be able to use the three constraints on your solution vector (unit vector, specified angle with a vector, specified angle with another vector) to write down three equations in the components of your solution vector. Then, solve.

 

[gunnar]

O.K I figured out that the angle between the given vectors is 60 degrees or pi/3

The angle between the unit vectors and each of the given vectors is supposed to be pi/3 also. So I used the dot product to calculate

c=sqrt(2) since the unitvector has length 1 and both the given vectors have the length sqrt(2)

Don't seem to be able to get the right answear.
The correct answear is (1/sqrt(2), 1/sqrt(2), 0) only one vector.

How to reach that conclusion I have no idea

 

[da_willem]

Lets call the components of the vector you seek (a,b,c). Now there are three restriction on these three numbers. So when you write down the tree restrictions Hurkyl gave in terms of a,b and c you have three equations with three unknowns which you can solve. Can you translate the restrictions from words to algebra?