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 (a_1, a_2, a₃) ∈ R³, aiˆ + a ˆj + akˆ is called the position vector of the point (a_1, a_2, 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