RESOLUTION OF A VECTOR IN A GIVEN BASIS
Consider two non-collinear vectors  and  ; as discussed earlier, these will form a basis of the plane in which they lie. Any vector  in the plane of and can be expressed as a linear combination of and :
=  + 
=l + m
for some l,m ÎR
The vectors and are called the components of the vector along the basis formed by and . This
is also stated by saying that the vector when resolved along the basis formed by and , gives the components
and
. Also, as discussed earlier, the resolution of any vector along a given basis will be unique.
We can extend this to the three dimensional case: an arbitrary vector can be resolved along the basis formed by any three non-coplanar vectors, giving us three orresponding components. Refer to Fig - 20 for a visual picture.
RECTANGULAR RESOLUTION
Let us select as the basis for a plane, a pair of unit vector i and j perpendicular to each other.
Any vector in this basis can be written as
where x and y are referred to as the x and y components of .
For 3-D space, we select three unit vectors i, j and k each perpendicular to the other two.
In this case, any vector will have three corresponding components, generally denoted by x, y and z. We thus
have
= xi + yj + zk
The basis ( i, j ) for the two dimensional case and ( i, j, k ) for the three-dimensional case are referred to as rectangular basis and are extremely convenient to work with. Unless otherwise stated, we.ll always be using a rectangular basis from now on. Also, we.ll always be implicitly assuming that we.re working in three dimensions since that automatically covers the two dimensional case.
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