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Question

The below figure shows a PQR triangle. M and N are respectively midpoints for PQ and PR. Given #vec (PR)# = 4 #ul x # and #vec (PM) # = #ul y #, express, in terms of #ul x # and #ul y #.

(a) #vec (QR) #,

(b) #vec (MN) #

Show that MN is parallel to QR.

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Answer 1 · 2018-04-25T11:19:53

See below.

If #M# is the midpoint of #PQ#:

Then #PM=MQ#

And:

#vec(PQ)=vec(PM)+vec(MQ)=bby+bby=2bby#

#vec(QP)=-vec(PQ)=-2bby#

#vec(PR)=4bbx#

#vec(QR)=vec(QP)+vec(PR)=-2bby+4bbx#

If #N# is the midpoint of #PR#

Then:

#PN=NR#

And:

#PR=PN+NR#

#vec(PR)=4bbx=>vec(PN)=vec(NR)=2bbx#

#vec(MN)=-vec(PM)+vec(PN)=-bby+2bbx#

Proof that #MN and QR# are parallel:

#vec(MN)=-bby+2bbx#

#vec(QR)=-2bby+4bbx=2(-bby+2bbx)=>vec(QR)=2vec(MN)#

Two vectors are parallel if one vector is a scalar multiple of the other vector.