Establish the following vector inequalities geometrically or otherwise:

   

Answer –

Let two vectors and represent two adjacent sides of a parallelogram PQRS, as show in the diagram below :

Here,

and

Also,

We know that for a triangle, each side is always smaller than the sum of other two sides. So, we can write

PR < QR + SR

   

Now, if both the said vectors act in a straight line then we can write,

 

   

Upon combining (1) and (2), we get the required result as,