Vector Properties: Divide by Direction Impossible

[Prashasti]

It is well known that a vector can't be divided by a vector, as a direction can't be divided by a direction. Keeping this in mind, I used the equation, v→ = u→+a→t, and wrote it as t = v→ -u→/a→. Now, isn't it wrong to write the equation like this? As , in it, a vector, that is v→ -u→ is being divided by another, (i.e. a→)?

 

[Hardik Batra]

This shows only equation representation. Actually you can't perform any operation like dividing.

 

[jtbell]

Consider
$$\vec v = \vec u + \vec a t\\
\vec v - \vec u = \vec a t$$

If t > 0, then the resultant of $\vec v - \vec u$ has the same direction as $\vec a$. If t < 0, then the resultant of $\vec v - \vec u$ has the opposite direction as $\vec a$.

If you know that two vectors $\vec a$ and $\vec b$ are along the same line (i.e. one equals the other multiplied by a scalar), then you can divide one by the other to get the scalar.

Otherwise, then you indeed cannot divide them.

 

[mathman]

If you are interested in the scalar, then all you need to divide is any (non-zero) component into its corresponding component.

In fact when you are dividing one vector by another, you are in essence carrying out three divisions instead of one to get the same result.