Product used in equations of motion [closed]

Question

Suppose $\vec{v}$, $\vec{u}$, $\vec{x}$, $\vec{a}$ and $t$ are final velocity, initial velocity, displacement and acceleration vectors and time.

Here are three equations of motion:

$\vec{v}$ = $\vec{u}$ + $\vec{a}$$t$

$\vec{x}$ = $\vec{u}$$t$ + $\frac 12$$\vec{a}$$t^2$

$\vec{v}^2$ = $\vec{u}^2$ + $2\vec{a}$$\vec{x}$

I wanted to know:

(1)What type of vector product is used in these equations?

(2)How can we obtain time from first equation if $\vec{v}$, $\vec{u}$,$\vec{a}$ are given as division of vectors in not allowed?

Answer 1

(1) The vector product in the third equation is the dot product, which you can see as follows:

$ |\vec v|^2 = \vec v . \vec v = (\vec u + \vec a t) . (\vec u + \vec a t) \\ = \vec u . \vec u + 2 \vec u . \vec a t + \vec a . \vec a t^2 \\ = \vec u . \vec u + 2\vec a .(\vec u t + \frac 1 2 \vec a t^2) \\ = \vec u . \vec u + 2\vec a .\vec x = |\vec u|^2 + 2\vec a .\vec x$

(2) If $\vec v - \vec u$ is a scalar multiple of $\vec a$ then you are fine. If $\vec v - \vec u$ is not a scalar multiple of $\vec a$ then there is an error in one the vectors you have been given.