I really need some assistance with the following problem:
"Find all unit vectors in the plane determined by u = (3,0,1) and v = (1,-1,1) that are perpendicular to the vector w = (1,2,0)".
The problem for me is that I donīt really know how to approach planes in general. For example, how do I calculate what plane is determined by the two vectors u and v?
What would u and v look like if they were serparate planes?
If someone know a website that explains the basics of vector planes I would appreciate the adress.
Thanks in advance
It may help if you think first of the unit vectors and that are used to define the plane. Any vector in this plane can be written , representing as it does, displacements and parallel to and respectively. ( and are, of course, any scalar numbers.)
In the same way, any vector of the form will lie in the plane defined by and . We may think of the grid of unit squares defined by and as having been replaced by a grid of parallelograms defined by and .
So if and , any vector lying in the plane determined by and will be of the form .
Now two vectors are perpendicular if their dot (scalar) product is zero. So this vector is perpendicular to , if:
So the vector will be of the form
All that remains is to find unit vectors in this form. So that will be where
So I reckon there's our answer: the unit vectors are