If #a# and #b# are unit vectors and #theta# is the angle between them, express #abs(a-b)# in terms of #theta# ?

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Answer 1 · 2017-08-24T09:48:05

# |veca-vecb|=2sin(theta/2).#

Suppose that, #veca, and vecb# are such unit vectors, that,

#hat{((veca, vecb))}=theta, theta in [0,pi].#

#because veca, &, vecb" are unit vectors, ":. |veca|=|vecb|=1....(0).#

We know, #(1): veca*vecb=|veca|*|vecb|*costheta,# and,

#(2): |vecx|^2=vecx*vecx.#

# :., |veca-vecb|^2=(veca-vecb)*(veca-vecb)............[because, (2)],#

#=veca*(veca-vecb)-vecb*(veca-vecb),#

#=veca*veca-veca*vecb-vecb*veca+vecb*vecb,#

#=|veca|^2-2veca*vecb+|vecb|^2...[because, veca*vecb=vecb*veca],#

#=1-2|veca|*|vecb|*costheta+1......[because, (0), and, (1)],#

#=2-2*1*1costheta...........[because, (0)],#

#=2(1-costheta)=2(2sin^2(theta/2)).#

# rArr |veca-vecb|=2sin(theta/2).#

Enjoy Maths.!