Colt Steele
Posted on Saturday, 06 December, 2003 - 03:44 pm:
1) find the components of the two unit vectors
w=(w1,w2,w3) which make an angle of 60 degrees with the vector u=(1,0,-1) and an angle of 45 degrees with the vector v=(1,-2,-2).
2) show that these two unit vectors are perpendicular.
Arne Smeets
Posted on Saturday, 06 December, 2003 - 05:13 pm:
Do you know the dot product?

I'll denote vectors in bold. It says that v.w=|v||w|α, where α is the angle between these vectors. Also, if a=(x,y,z) and b=(x',y',z') then a.b=xx'+yy'+zz'. Try to use this.

Chris Tynan
Posted on Saturday, 06 December, 2003 - 05:26 pm:
I think you mean v.w=|v||w|cosα where α is the angle between them.
Arne Smeets
Posted on Saturday, 06 December, 2003 - 07:23 pm:
Of course I did.
Colt Steele
Posted on Sunday, 07 December, 2003 - 01:23 am:
I still can't solve it. There are too many variables. I don't understand.
Arne Smeets
Posted on Sunday, 07 December, 2003 - 10:12 am:
Ok. We know that LaTeX Image = 1.

We have LaTeX Image.
Similarly, LaTeX Image.
Finally, we have LaTeX Image.
Now you only have to solve this system.
Arne Smeets
Posted on Sunday, 07 December, 2003 - 10:35 am:
So, we have to solve the system

LaTeX Image
LaTeX Image
LaTeX Image.

Subtract the first two equations to get
LaTeX Image.

So we get LaTeX Image.

Hence LaTeX Image.
So we then get LaTeX Image
Hence LaTeX Image or LaTeX Image.

The corresponding vectors are LaTeX Image and LaTeX Image.

It is easy to check that they are orthogonal:
LaTeX Image