Question

Can two vectors, of unequal magnitude, add up to give the zero vector? Can three unequal vectors? Under what conditions?

Short answer

Two vectors of unequal magnitude cannot add up to give a resultant zero vector. In the case of three vectors of unequal magnitude, the vector sum will be equal to zero if the vector sum of any two vectors is equal to the negative of the third vector.

Step-by-step solution

Step1. Explanation for zero vector

A zero vector or null vector is defined as a vector of zero magnitude, which implies that all components of that vector are zero.

Step 2. Checking for the case of two vectors of unequal magnitude

Let two vectors ${\overrightarrow{V}}_1$and ${\overrightarrow{V}}_2$have magnitudes $V_1$and $V_2$, respectively, such that $V_1\ne V_2$. For now, let us assume that the vector sum of these two vectors is zero. Thus,

${\overrightarrow{V}}_1+{\overrightarrow{V}}_2=0$

Adding vector $-{\overrightarrow{V}}_2$on both sides, you get:

${\overrightarrow{V}}_1=-{\overrightarrow{V}}_2$

This is only possible when $V_1\ne V_2$. Thus, for two vectors of unequal magnitude, their vector sum cannot be a zero vector.

Step 3. Checking for the case of three vectors of unequal magnitude

Consider three vectors ${\overrightarrow{V}}_1$, ${\overrightarrow{V}}_2$, and ${\overrightarrow{V}}_3$having magnitudes $V_1$, $V_2$, and $V_3$, respectively, such that $V_1\ne V_2\ne V_3$. For now, let us assume that the vector sum of these three vectors is zero. Thus,

${\overrightarrow{V}}_1+{\overrightarrow{V}}_2+{\overrightarrow{V}}_3=0$

Adding vector $-{\overrightarrow{V}}_3$on both the sides, you get:

${\overrightarrow{V}}_1+{\overrightarrow{V}}_2=-{\overrightarrow{V}}_3$

Thus, if out of three, the sum of any two vectors is equal to the negative of the third vector, then the vector sum of all three vectors will be zero, even when all the vectors are different.