Vectors

If an object is travelling in a three dimensional space, wherein the directions are defined as x, y, and z axis.

Position Vector

Then the position of the object relative to the origin at any given time $t$ is,

$r(t)=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}$

$\hat{i},\hat{j},\hat{k}$ are called unit vectors, that is it defines 1 unit distance in each directions. The term ‘vector’ here signifies that a quantity is directional in nature. Compared ‘scaler’ in a non-directional nature of a quantity.

To find the distance of the object from origin, we need to apply Pythagoras Theorem to each individual planar region, on computation we get a following formula for distance in a 3D space:

$|r|=\sqrt{x^2+y^2+z^2}$

Velocity Vector

The first derivative of position vector will give us the coordinates of velocity vectors in each direction.

$v(t)=\frac{dr(t)}{dt}$

Acceleration Vector

The second derivation of position vector,

$a(t)=\frac{d^{2}r(t)}{d{t}^2}$