IIT JEE , AIEEE Forum - Mathematics , Vectors

cauchy089

If ABCD is quadrilateral and E and F are the mid points of AC and BD respectively,

Then AB+AD+CB+CD , is equal to 4 EF

elessar_iitkgp

Draw the diagram. Take any arbitrary point as the origin. Let the position vectors of A, B, C, D be a, b, c, d respectively.
Position vector of E, e = (a + c)/2
Position vector of F, f = (b + d)/2
AB + AD + CB + CD = (b - a) + (d - a) + (b - c) + (d - c) = 2(b+d-a-c)
= 4[(b+d)/2 - (a+c)/2] = 4(f - e) = 4EF

magiclko

let origin be at D, and then a, b, c, e and f be the vectors to points A,B,C,E and F resp.

thrfore f = b / 2

e = (a+c) /2

thrfore, AB+AD+CB+CD = (b-a) + (0-a) + (b-c) + (0-c)

= 2 (b-a-c)

= 4 [b /2 - (a+c)/2]

= 4 (f - e)

= 4EF Proved

elessar_iitkgp

Good solution Magiclko.
Cauchy, you have two solutions: One with an arbitrary origin and one with origin at one of the given points.
Understand this, a vector result is independent of the origin unless the origin itself moves.

futureiitian

Hi I find vector topic difficult. Can some one suggest me a good book to study vector for iitjee? Thanks

Ask & Discuss Questions with Community & Experts