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.

 

[pmacias]

Yes, it is incorrect to divide a vector by another vector. In mathematics, division is defined as the inverse operation of multiplication, but for vectors, there is no defined operation for division. Vectors are quantities that have both magnitude and direction, and dividing by a vector would not make sense in this context.

In the equation t = v→ -u→/a→, the vector v→ -u→ represents the displacement between two points, and a→ represents the acceleration. It is important to note that the division symbol in this equation is not meant to be interpreted as a mathematical operation, but rather as a way to represent the relationship between these quantities.

Furthermore, the equation v→ = u→+a→t is a simplified version of the kinematic equation for constant acceleration, which is derived from the fundamental equations of motion. This equation is valid for one-dimensional motion and cannot be used in a situation where the acceleration is not constant.

In conclusion, it is incorrect to divide a vector by another vector. Rather than trying to divide by direction, it is important to understand and use the appropriate equations and concepts for vector operations in order to accurately represent physical quantities.