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.

Parallelogram - Vector Geometry


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