Mathematics : Geometric introduction to vectors

**Geometric
introduction to vectors**

A vector * *is represented as a
directed straight line segment in a 3-dimensional space **R**^{3} , with an
initial point *A *= (*a*_{1}, *a*_{2}, *a*_{3}) âˆˆ **R**^{3} and an end point *B
*= (*b*_{1},
*b*_{2}, *b*_{3}* *)âˆˆ **R**^{3} , and it is
denoted by . The length of the line segment *AB
*is the magnitude of the vector * *and the direction from *A
*to *B *is the direction of the vector . Hereafter,
a vector will be interchangeably denoted by * *or . Two vectors * *and in **R**^{3} are said to be equal if
and only if the length *AB *is equal to the length *CD *and the
direction from *A *to *B *is parallel to the direction from *C *to
*D *. If * *and * *are equal, we write * *= * *, and * *is called a **translate **of .

It is easy to observe that every vector * *can be
translated to anywhere in **R**^{3}
, equal to a vector with initial point *U *âˆˆ **R**^{3} and end point *V *âˆˆ **R** 3 such that * *= * *. In
particular, if *O *is the origin of **R**^{3} , then a point *P *âˆˆ **R**^{3} can
be found such that * *= . The vector * *is called the **position vector **of the point *P *. Moreover, we observe
that given any vector , there exists a unique point *P *âˆˆ **R**^{3} such that the
position vector * *of *P *is equal to * *.
A vector * *is said to be the **zero vector **if the initial point *A
*is the same as the end point *B *. We use the standard notations *i*Ë†,
Ë†*j*, *k*Ë† and to denote the position vectors of the points
(1, 0, 0),(0,1, 0),(0, 0,1), and (0, 0, 0), respectively. For a given point (*a*_{1,} *a*_{2,} *a*_{3})
âˆˆ **R**^{3}, *ai*Ë† + *a* Ë†*j* + *ak*Ë†
is called the position vector of the point (*a*_{1,} *a*_{2,}
*a*_{3}) which is the directed
straight line segment with initial point (0, 0, 0) and end point (*a*_{1}
, *a*_{2} , *a*_{3} ) . All real numbers are called
scalars.

Given a vector * *, the **length of the vector
|****| **is
calculated by

where *A *is (*a*_{1} , *a*_{2} ,
*a*_{3} ) and *B *is (*b*_{1} , *b*_{2}
, *b*_{3} ). In particular, if a vector is the position vector * *of* (b*_{1} , *b*_{2} , *b*_{3} ),
then its length is

A vector having length 1 is called a **unit vector**. We the notation *u*Ë†
, for a unit vector. Note that *i*Ë†, Ë†*j *, and *k*Ë† use
are unit vectors and is the unique vector with length 0 . The
direction of is specified according to the context.

The **addition **and **scalar multiplication **on vectors in 3-dimensional space are defined
by

To see the geometric interpretation of * *+ ,
let * *and , denote the position vectors of *A *=
(*a*_{1} , *a*_{2} , *a*_{3} ) and *B *=
(*b*_{1} , *b*_{2} , *b*_{3} ) ,
respectively. Translate the position vector to the vector
with initial point as *A *and end point as *C *= (*c*_{1}
, *c*_{2} , *c*_{3} ) , for a suitable (*c*_{1}* *, *c*_{2}* *, *c*_{3}* *) âˆˆ **R**^{3}. See the Fig (6.2). Then, the position vector * *of the point (*c*_{1} , *c*_{2} , *c*_{3}
) is equal to * *+ .

The vector *Î±** *is another vector parallel to * *and
its length is magnified (if *Î± *> 1) or contracted (if 0 < *Î±
*< 1) . If *Î± *< 0 , then *Î± ** *is a vector
whose magnitude is | *Î± *| times that of * *and
direction opposite to that of . In particular, if *Î± *=
âˆ’1, then *Î±*= âˆ’is the vector with same length
and direction opposite to
that of . See Fig. 6.3

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Geometric introduction to vectors |

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