Question: Prove that $\vec{PQ} + \vec{QR} + \vec{RS} + \vec{SP} = 0$ in the given quadrilateral PQRS.

Solution:

To prove: $\vec{PQ} + \vec{QR} + \vec{RS} + \vec{SP} = 0$

Construction: Join P and R.



Proof:

Using ∆ law of vector addition,

$\vec{PR} = \vec{PQ} + \vec{QR}$ - (i)

Also,

$\vec{RP} = \vec{RS} + \vec{SP}$ - (ii)

Taking LHS,

$= \vec{PQ} + \vec{QR} + \vec{RS} + \vec{SP}$

[Put value of $\vec{PR} = \vec{PQ} + \vec{QR}$ from equation (i) ]

$= (\vec{PR}) + \vec{RS} + \vec{SP}$

[Put value of $\vec{RP} = \vec{RS} + \vec{SP}$ from equation (ii) ]

$= \vec{PR} + \vec{RP}$

$= \vec{PR} - \vec{PR}$

$= 0$

RHS
#proved