Vector Representation of Motion:
If the motion of a particle is described by a position vector, then
r (t ) = x (t ) i¯ + y (t ) j¯ + z (t ) k¯ where i¯ , j¯ and k¯ are unit vectors along x, y and z-axes, respectively.
The direction cosines of this vector are cos α, cos β, cos γ. We have
r (t ) = x (t ) i¯ + y (t ) j¯ + z (t ) k¯
Then velocity
v = dr/ dt = dx/ dt i¯ +dy/dt j¯ + dz/dt k¯
∴ v = vx i¯ + v y j¯ + vz k¯
and
![1800_Vector Representation of Motion.png](https://www.expertsmind.com/CMSImages/1800_Vector%20Representation%20of%20Motion.png)
and
cos α = vx/v , cos β = v y/v , cos γ = vz/v
Likewise acceleration of the particle shall be obtained by
a = dv/dt = d2r/dt = d2x/dt i¯ + d2y/dt j¯ + d2z/dt k¯
or a = ax i¯ + a y j¯ + az k¯
![15_Vector Representation of Motion1.png](https://www.expertsmind.com/CMSImages/15_Vector%20Representation%20of%20Motion1.png)
and
ax/a = cos α, ay/a= cos β, az /a = cos γ