1.
Explain resolution of vector in two dimension.
Answers
Answer:
The resolution of a vector into two mutually perpendicular vectors is called the rectangular resolution of vector in a plane or two dimensions. be the unit vectors along X-axis and Y-axis respectively figure.
Answer:
Resolution Of Vectors
Go back to 'Vectors and 3-D Geometry'
RESOLUTION OF A VECTOR IN A GIVEN BASIS
Consider two non-collinear vectors
→
a
and
→
b
; as discussed earlier, these will form a basis of the plane in which they lie. Any vector
→
r
in the plane of
→
a
and
→
b
can be expressed as a linear combination of
→
a
and
→
b
:
The vectors
−−→
O
A
and
−−→
O
B
are called the components of the vector
→
r
along the basis formed by
→
a
and
→
b
. This is also stated by saying that the vector
when resolved along the basis formed
b
, gives the components
. 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 corresponding 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
and
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 are
For 3-D space, we select three unit vectors
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
The basis (
) for the two dimensional case and
) 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.