Vectors- Types And Components

Vectors- Types And Components

Displacement and Displacement vector   
    Displacement is a shortest distance between two points. It is a vector quantity.
    The position of the point Q with reference to the origin is represented by the position vector \( \mathop {r_2 }\limits^ \to \). Let the coordinates of the point Q are (x₂, y₂)

Similarly  represent by a position vector \( \mathop {r_1 }\limits^ \to \), let the coordinates of the point P are (x₁, y_A) As the displacement vector is the difference of two position vectors \( \mathop {r_1 }\limits^ \to \) = x₁\( \mathop i\limits^ \wedge \) + y₁\( \mathop j\limits^ \wedge \) and  \( \mathop {r_2 }\limits^ \to \)= x₂\( \mathop i\limits^ \wedge \) + y₂\( \mathop j\limits^ \wedge \), where \( \mathop i\limits^ \wedge \), \( \mathop j\limits^ \wedge \)  are unit vectors along X, Y axis respectively.
    Thus, the displacement vector \( \Delta \mathop r\limits^ \to = \text{ }\mathop {r_2 }\limits^ \to - \mathop {r_1 }\limits^ \to \) =  (x₂ – x_A)\( \mathop i\limits^ \wedge \) + (y₂ – y_A)\( \mathop j\limits^ \wedge \)
Basic addition of vectors
    To add two vectors, one vector is drawn to scale and the tail of second vector is made to coincide with the head of first vector without changing its orientation.

Then the line joining the tail of first vector and the head of last vector represents the resultant both in magnitude and direction.

Laws of Vector Addition:
Vector addition follows commutative,associative and distributive laws.
a) Commutative law :\( \overline A + \overline B = \overline B + \overline A \) 
b) Associative law : 
c) Distributive law :  where m is a scalar

Subtraction of a vector from another vector:
    The subtraction of one vector from the other is a special case of addition of two vectors in which one vector is compounded with the negative of another. \( \overline a - \overline b = \overline a + \left( { - \overline b } \right) \)   
Note: If \( \overrightarrow A = A_x \hat i + A_y \hat j + A_z \hat k \) and \( \overrightarrow B = B_x \hat i + B_y \hat j + B_z \hat k \) then
A)  \( \overrightarrow A + \overrightarrow B = \left( {A_x + B_x } \right)\hat i + \left( {A_y + B_y } \right)\hat j + \left( {A_z + B_z } \right)\hat k \)   
B)  \( \overrightarrow A - \overrightarrow B = \left( {A_x - B_x } \right)\hat i + \left( {A_y - B_y } \right)\hat j + \left( {A_z - B_z } \right)\hat k \)  
C)  If they are parallel, then \( \frac{{A_x }} {{B_x }} = \frac{{A_y }} {{B_y }} = \frac{{A_z }} {{B_z }} \) 

Product of a vector with a scalar : 
a)  \( \overrightarrow A = s\overrightarrow B \) When a vector is multiplied by a scalar, the resultant is also a vector. 
b)  \( \overrightarrow A = s\overrightarrow B \) , If s is positive scalar then direction of \( \overrightarrow A \) is the same as that of \( \overrightarrow B\). 
     If s is negative then direction of \( \overrightarrow A \) is opposite to that of \( \overrightarrow B \) 
c)  Vector multiplication obeys commutative law when multiplied by a scalar. 
     i.e.,\( s\,\overrightarrow A = \overrightarrow A \,s \)  where s is a scalar
d)  Vector multiplication obeys associative law when multiplied by scalar.
     i.e.\( m\left( {n\overrightarrow A } \right) = mn\,\overrightarrow A \)  (m, n are scalars)
e)  Vector multiplication obeys distributive law when multiplied by a scalar. 
      i.e.\( s\left( {\overrightarrow A + \overrightarrow B } \right) = s\overrightarrow A + s\overrightarrow B \)  (s is a scalar).
Note:
A) A vector can be added to another vector or can be subtracted from another vector, the result also a vector
B) A vector can be multiplied with a scalar (or) with reciprocal of scalar, the result also vector.
C) Vector, vector division is not possible.
D) If a vector is represented by \( x\hat i + y\hat j + z\hat k \). Its length, in X – Y plane is \( \sqrt {x^2 + y^2 } \), in Y – Z plane is \( \sqrt {y^2 + z^2 } \), in X – Z plane is \( \sqrt {x^2 + z^2 } \)