12 - University Physics with Modern Physics

Question

Can you find two vectors with different lengths that have a vector sum of zero? What length restrictions are required for three vectors to have a vector sum of zero? Explain.

Step-by-Step Solution

Answer

No it is not possible for the vectors of different length to have a vector sum of zero.

Vector

Vector is a way of representing a quantity which have magnitiude as well as direction.

A quantity is defined by its magnitude as well as the direction in which the magnitude is acting. For example force, displacement etc.

 

Sum of two forces

Consider two forces F₁ and F_2. such that the sum of them is zero. It is only possible when both the force are equal to each other and opposite in direction.

Hence we get,

 $$\begin{gathered} F_1=-F_2 \\ {F}_1+F_2=0 \end{gathered}$$ 

Hence it is clear that we cannot find two vectors having different lengths (magnitude) such that their sum is zero.

Sum of three forces

For the three force system, it is also not possible to get the three vectors of different lengths having sum zero. 

 $$\begin{gathered} F_1=-\left(F_2+F_3\right) \\ {F}_1+F_2+F_3=0 \end{gathered}$$

 Therefore, we can conclude that it is not possible for the vectors of different length that have vectors sum zero.