# 3 particles start from origin with velocity u_1,u_2 and u_3.They are moving with constant velocities.Find the relation between u_1,u_2 and u_3 so that all 3 particles lie on the same plane?

Mar 19, 2018

$\left({\vec{u}}_{1} - {\vec{u}}_{2}\right) \times \left({\vec{u}}_{1} - {\vec{u}}_{3}\right) \ne \vec{0}$

#### Explanation:

The path for the particles is given by

${p}_{i} = {p}_{0} + t {\vec{u}}_{i}$

with ${p}_{0} = \left(0 , 0 , 0\right)$

The condition for non-alignment is

$\left({p}_{1} - {p}_{2}\right) \times \left({p}_{1} - {p}_{3}\right) \ne \vec{0}$ or

$\left(t {\vec{u}}_{1} - t {\vec{u}}_{2}\right) \times \left(t {\vec{u}}_{1} - t {\vec{u}}_{3}\right) \ne \vec{0}$

or

$\left({\vec{u}}_{1} - {\vec{u}}_{2}\right) \times \left({\vec{u}}_{1} - {\vec{u}}_{3}\right) \ne \vec{0}$