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Resolution Of Vectors


RESOLUTION OF A VECTOR IN A GIVEN BASIS

Consider two non-collinear vectors \(\vec a\,\,{\text{and}}\,\,\vec b\); as discussed earlier, these will form a basis of the plane in which they lie. Any vector  \(\vec r\) in the plane of \(\vec a\,\,{\text{and}}\,\,\vec b\) can be expressed as a linear combination of \(\vec a\,\,{\text{and}}\,\,\vec b\):

The vectors \(\overrightarrow {OA} \,\,\,{\text{and}}\,\,\overrightarrow {OB} \) are called the components of the vector \(\vec r\) along the basis formed by \(\vec a\,\,{\text{and}}\,\,\vec b\) . This is also stated by saying that the vector \(\vec r\) when resolved along the basis formed by \(\vec a\,\,{\text{and}}\,\,\vec b\) , gives the components  \(\overrightarrow {OA} \,\,\,{\text{and}}\,\,\overrightarrow {OB} \) . 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 \(\hat i\,\,{\text{and}}\,\,\hat j\) perpendicular to each other.

Any vector \(\vec r\) in this basis can be written as

\[\begin{align}&\vec r = \overrightarrow {OA}  + \overrightarrow {OB}  \hfill \\&\;= \left( {\left| {\vec r} \right|\cos \theta } \right)\hat i + \left( {\left| {\vec r} \right|\sin \theta } \right)\hat j \hfill \\&\;= x\hat i + y\hat j \hfill \\ 
\end{align} \]

where x and y are referred to as the x and y components of  \(\vec r\).

For 3-D space, we select three unit vectors \(\hat i,\hat j\,\,{\text{and}}\,\,\hat k\) each perpendicular to the other two.

In this case, any vector \(\vec r\) will have three corresponding components, generally denoted by x, y and z. We thus have

\[\vec r = x\hat i + y\hat j + z\hat k\]

The basis (\(\hat i,\hat j\)) for the two dimensional case and (\(\hat i,\hat j,\,\,\hat k\)) 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.

Download SOLVED Practice Questions of Resolution Of Vectors for FREE
Vectors
grade 11 | Questions Set 1
Vectors
grade 11 | Answers Set 1
Vectors
grade 11 | Questions Set 2
Vectors
grade 11 | Answers Set 2
Three Dimensional Geometry
grade 11 | Questions Set 2
Three Dimensional Geometry
grade 11 | Answers Set 2
Three Dimensional Geometry
grade 11 | Questions Set 1
Three Dimensional Geometry
grade 11 | Answers Set 1
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