Question

prove that the sum of a vector is represented by the sides of a closed polygon taken order is a zero vector​

Answer 1

Lets take an example of a hexagon, and according to your question, vectors a1, a2, a3, a4, a5, a6 make the side of the hexagon in the order as shown in the figure.

Now, from triangle law of vector addition,

a1 + a2 = b1

b1 + a3 = b2

b2 + a4 = b3

b3 + a5 = b4

Also, b4 = -a6

So, b4 + a6 = 0

So, the vectors added in an order gave a null vector.

Similarly,If a set of vectors taken in a given order gives a closed polygon, then the resultant of these vectors, will be a null vector.

Hope it helps.