Resolution Of Vectors Into Components And Types Of Vectors

Resolution Of Vectors Into Components And Types Of Vectors

Components of a vector
    Any vector can be expressed as the vector sum of some other vectors known as component vectors
Eg:\( \overline {AB} = \overline {AP} + \overline {PQ} + \overline {QR} + \overline {RB} \)  
The maximum number of vector components that a given vector can be resolved into is infinite.

Rectangular components :
    If a (plane) vector is resolved into two mutually perpendicular component vectors then these component vectors are called as the rectangular component vectors of that vector in 2 D coordinate system.
    If a (spatial) vector is resolved into three mutually perpendicular component vectors then these component vectors are called as the rectangular component vectors of that vector in 3D coordinate system. 
    The advantage of splitting up a vector into rectangular components is we can find the magnitude of the vector by simply finding the square root of sum of the squares of the components.  

RESOLUTION OF A VECTOR:
1.  A vector \(\overline{R} \) makes an angle \( \theta \), with X axis as shown, it can be resolved into two mutually perpendicular components R_x=OA and R_y=OB.    
2. The projection of  along x-axis is OA from right angled triangle OPA, \( \cos \theta = \frac{{OA}} {{OP}} = \frac{{R_x }} {R} \Rightarrow R_x = R\cos \theta \)
    Here \( \text{R}_\text{x} \text{ = R}\text{.cos} \) \(\theta\)is called horizontal component 

3. The projection of \(\overline{R}\) along y-axis is OB 
    from right angled triangle OPA, \( \sin \theta = \frac{{AP}} {{OP}} = \frac{{OB}} {{OP}} = \frac{{R_y }} {R} \Rightarrow R_y = R\sin \theta \) 
    Here \( \text{R}_\text{y} \text{ = R}\text{.sin} \)\(\theta\) called vertical component.
4.  Magnitude of the resultant \( \left| {\overline{ R}} \right|\text{ = }\sqrt {\text{R}_\text{x} ^\text{2} \text{ + R}_\text{y} ^\text{2} } \)                 
5. Here \( \text{R}_\text{x} \text{ = R}{.cos\theta } \) and \( \text{R}_\text{y} \text{ = R}{.sin\theta } \) are called as components and \( \overrightarrow {R_x } = R\cos \theta \overrightarrow {i,} \overrightarrow {\,\,R_y } = R\sin \theta \overrightarrow j \) are called vector components. 
    Vector \( \overrightarrow R = \overrightarrow {R_x } + \overrightarrow {\,\,R_y } = R\cos \theta \overrightarrow {i,} + R\sin \theta \overrightarrow j \) 
6. Direction of the resultant with x-axis \( {\theta = tan}^{\text{ - 1}} \left( {\frac{{\text{R}_\text{Y} }} {{\text{R}_\text{X} }}} \right) \)    
Note: In the above if \( \overrightarrow R \) makes angle \( \alpha ,\beta \) with x and y axis respectively then 
\( \cos \alpha = \cos \theta = \frac{{R_x }} {{\sqrt {R_x^2 + R_y^2 } }},\cos \beta = \cos (90 - \theta ) = \frac{{R_y }} {{\sqrt {R_x^2 + R_y^2 } }} \)