Can the magnitude of the difference between two vectors ever be greater than the magnitude of either vector? Can it be greater than the magnitude of their sum? Give examples.

Deepak Patra
8 years ago
Consider a two dimensional vector with component along the unit vector in x-direction , along the unit vector in y-direction respectively.
Mathematically, the vector can be represented as:

The magnitude of the vector  is given as:

Consider another vector with component along the unit vector in x-direction,  along the unit vector in y-direction respectively.
Mathematically, the vector can be represented as:

The magnitude of the vector is given as:

The difference of vectors and is given as:



The magnitude of the vector can be given as:
 …… (1)
The magnitude of difference between the two vector given in the equation above depends on the value of and respectively. In a situation, where the sign of is opposite to that with the sign of or the sign of is opposite to the sign of , the magnitude of vector can become greater to the magnitude of either vector or vector 
For example,
Let us assume that vector  is given as:

Therefore the magnitude of vector is:


Let us assume that the vector is given as:

The magnitude of vector  is:


Now we calculate the magnitude of vector , which is:



Therefore the magnitude of vector is greater than the magnitude of vector and vector.
The sum of vector  and vector is given as:


The magnitude of the vector can be given as:
 …… (2)
If the magnitude of vector is greater than the magnitude of vector , we have

Therefore, if the vectors and are such that the condition is fulfilled, the magnitude of vector will be greater than the magnitude of vector .
For example,
Let us assume that vector  is given as:

Let us assume that the vector is given as:

Now, we check if the condition is fulfilled by substituting the components of vectors as:




Substituting the values in the condition, we have
(5) (1) <-(-6)(2)
5 <12
Thus, our condition holds for the given vectors.
Now, we calculate the magnitude of vector and vector as:
The magnitude of vector  is:

The magnitude of vector is:

Therefore the magnitude of vector is greater than the magnitude of vector , as we expected it to be.