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Chapter: 12th Mathematics : UNIT 6 : Applications of Vector Algebra

Geometric introduction to vectors

Mathematics : Geometric introduction to vectors

Geometric introduction to vectors

A vector  is represented as a directed straight line  segment in a 3-dimensional space R3 , with an  initial  point A  =  (a1, a2, a3) R3 and an end point  B  = (b1, b2, b3 ) R3 , 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 R3 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 R3 , equal to a vector with initial point U R3 and end point V R 3 such that   . In particular, if O is the origin of  R3 , then a point P R3 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 R3 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 iˆ, ˆj, kˆ  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 (a1, a2, a3) R3, aiˆ + a ˆj + akˆ is called the position vector of the point (a1, a2, a3) which is the directed straight line segment with initial point (0, 0, 0) and end point (a1 , a2 , a3 ) . All real numbers are called scalars.

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


where A is (a1 , a2 , a3 ) and B is (b1 , b2 , b3 ). In particular, if a vector is the position vector  of (b1 , b2 , b3 ), then its length is 


A vector having length 1 is called a unit vector. We the notation uˆ , for a unit vector. Note that iˆ, ˆj , and kˆ  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 = (a1 , a2 , a3 ) and B = (b1 , b2 , b3 ) , respectively. Translate the position vector  to the vector with initial point as A and end point as C = (c1 , c2 , c3 ) , for a suitable (c1 , c2 , c3 ) R3. See the  Fig (6.2). Then, the position vector  of the point (c1 , c2 , c3 ) is equal to  .

The vector α is another vector parallel to  and its length is magnified (if α > 1) or contracted  (if 0 < α < 1) . If α < 0 , then α  is a vector whose magnitude is | α | times that of  and direction opposite to that of . In particular, if α = −1, then α= −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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12th Mathematics : UNIT 6 : Applications of Vector Algebra


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