Parametric Equations and Position Vector Representation of Parabola

1. In Parametric Equations of Parabola, the Coordinates \(x\) and \(y\) (and \(z\) for Parabolas in 3 Dimensions) are given in terms of a Real Number Parameter which is generally denoted by \(t\).
2. The following equations represent the Parametric Equation of Parabola in 2D Space
   
   \(x=A_1t^2 \pm B_1t + C_1,\hspace{.5cm}y=A_2t^2 \pm B_2t + C_2\)   ...(1)
   
   The Parametric Equations of Parabola in 2D Space given above can also be written in form of Position Vector Equation \(\vec{P_2}\) as follows
   
   \(\vec{P_2} = \begin{bmatrix}A_1 \\ A_2\end{bmatrix}t^2 \pm \begin{bmatrix}B_1 \\ B_2\end{bmatrix}t + \begin{bmatrix}C_1 \\ C_2\end{bmatrix} = \vec{A}t^2 \pm \vec{B}t + \vec{C}t\)   ...(2)
   
   The following equations represent the Parametric Equation of Parabola in 3D Space
   
   \(x=A_1t^2 \pm B_1t + C_1,\hspace{.5cm}y=A_2t^2 \pm B_2t + C_2,\hspace{.5cm}z=A_3t^2 \pm B_3t + C_3\)   ...(3)
   
   The Parametric Equations of Parabola in 3D Space given above can also be written in form of Position Vector Equation \(\vec{P_3}\) as follows
   
   \(\vec{P_3} = \begin{bmatrix}A_1 \\ A_2 \\ A_3\end{bmatrix}t^2 \pm \begin{bmatrix}B_1 \\ B_2 \\ B_3\end{bmatrix}t + \begin{bmatrix}C_1 \\ C_2 \\ C_3\end{bmatrix} = \vec{A}t^2 \pm \vec{B}t + \vec{C}t\)   ...(4)
   
   In Position Vector Equations (2) and (4)
   
   Vector \(\vec{A}\) is the Acceleration Vector of the Parabola
   
   Vector \(\vec{C}\) is the Position Vector of any Point \(C\) on Parabola
   
   Vector \(\pm\hspace{1mm}\vec{B}\) is the Velocity Vector at Point \(C\) on Parabola
   
   For the Position Vectors \(\vec{P_2}\) and \(\vec{P_3}\) in equations (2) and (4) above to represent a Parabola the Vectors \(\vec{A}\) and \(\pm\hspace{1mm}\vec{B}\) Must Have Different Line Directions.
   
   Also, for the equations (1) and (3) to represent the Parametric Equations of Parabola atleast One of the Equations MUST HAVE the Quadratic Term for Parameter \(t\) (i.e atleast one of Co-efficients \(A_1\) or \(A_2\) (or \(A_3\) in case of Parabola in 3D Space) must be non-zero) AND atleast One of the Equations MUST HAVE the Linear Term for Parameter \(t\) (i.e atleast one of Co-efficients \(B_1\) or \(B_2\) (or \(B_3\) in case of Parabola in 3D Space) must be non-zero). The Constants \(C_1\), \(C_2\) and \(C_3\) can be zero or non-zero.
   
   This means in Position Vector Equations (2) and (4) the Acceleration Vector \(\vec{A}\) and the Velocity Vector \(\vec{B}\) cannot be Zero/Null Vector. However the Position Vector \(\vec{C}\) of the Point \(C\) can be Zero/Null Vector.
   
   Also for any given Parabola, the Acceleration Vector \(\vec{A}\) has the Same Constant Line Direction as the Axis of Parabola. However the Direction of Velocity Vector \(\pm\hspace{1mm}\vec{B}\) changes depending on the Point \(C\) chosen for/on the Parabola.
3. For Axis Aligned Parabolas in 2D Space, the Parametric Equations and the Position Vector Equations are given as follows
   
   \(x=\pm B_1t + C_1,\hspace{.5cm}y=A_2t^2 \pm B_2t + C_2\)   ...(5) (Parametric Equations for Parabolas having Directrix Parallel to \(X\) Axis)
   
   \(\vec{P_2} = \begin{bmatrix}0 \\ A_2\end{bmatrix}t^2 \pm \begin{bmatrix}B_1 \\ B_2\end{bmatrix}t + \begin{bmatrix}C_1 \\ C_2\end{bmatrix}\)   ...(6) (Position Vector Equation for Parabolas having Directrix Parallel to \(X\) Axis)
   
   
   \(x=A_1t^2 \pm B_1t + C_1,\hspace{.5cm}y=\pm B_2t + C_2\)   ...(7) (Parametric Equations for Parabolas having Directrix Parallel to \(Y\) Axis)
   
   \(\vec{P_2} = \begin{bmatrix}A_1 \\ 0\end{bmatrix}t^2 + \begin{bmatrix}B_1 \\ B_2\end{bmatrix}t + \begin{bmatrix}C_1 \\ C_2\end{bmatrix}\)   ...(8) (Position Vector Equation for Parabolas having Directrix Parallel to \(Y\) Axis)
   
   
   If the Point \(C\) on Parabola having Position Vector \(\vec{C}\) is chosen such that it is the Vertex of the Parabola having coordinates (\(x_v,y_v\)), then the Parametric Equations and the Position Vector Equations of the Axis Aligned Parabolas in 2D Space are given as follows
   
   \(x=\pm B_1t + C_1= \pm B_1t + x_v,\hspace{.5cm}y=A_2t^2 + C_2=A_2t^2 + y_v\)   ...(9) (Parametric Equations for Parabolas having Directrix Parallel to \(X\) Axis)
   
   \(\vec{P_2} = \begin{bmatrix}0 \\ A_2\end{bmatrix}t^2 \pm \begin{bmatrix}B_1 \\ 0\end{bmatrix}t + \begin{bmatrix}C_1 \\ C_2\end{bmatrix}= \begin{bmatrix}0 \\ A_2\end{bmatrix}t^2 \pm \begin{bmatrix}B_1 \\ 0\end{bmatrix}t + \begin{bmatrix}x_v \\ y_v\end{bmatrix}\)   ...(10) (Position Vector Equation for Parabolas having Directrix Parallel to \(X\) Axis)
   
   
   \(x=A_1t^2 + C_1=A_1t^2 + x_v,\hspace{.5cm}y=\pm B_2t + C_2= \pm B_2t + y_v\)   ...(11) (Parametric Equations for Parabolas having Directrix Parallel to \(Y\) Axis)
   
   \(\vec{P_2} = \begin{bmatrix}A_1 \\ 0\end{bmatrix}t^2 \pm \begin{bmatrix}0 \\ B_2\end{bmatrix}t + \begin{bmatrix}C_1 \\ C_2\end{bmatrix} = \begin{bmatrix}A_1 \\ 0\end{bmatrix}t^2 \pm \begin{bmatrix}0 \\ B_2\end{bmatrix}t + \begin{bmatrix}x_v \\ y_v\end{bmatrix}\)   ...(12) (Position Vector Equation for Parabolas having Directrix Parallel to \(Y\) Axis)
   
   
   This is because in such cases, the Velocity Vector \(\pm\hspace{1mm}\vec{B}\) is Perpendicular to the Acceleration Vector \(\vec{A}\).
   
   In such cases the Parametric Equations of Parabola are called the Standard Parametric Equations.