How do we prove the projection law using vectors?

2 Answers
Apr 23, 2018

Let vecaandvecb are two vectors inclined at an angle theta.

So

veca*vecb=absvecaabsvecbcostheta

Now vector projection of veca on to vecb will be
=absvecacostheta*"unit vector of "vecb

=absvecacostheta*vecb/(absvecb)
=absvecaabsvecbcostheta*vecb/(absvecb^2)

=(veca*vecb)vecb/(absvecb^2)

Apr 23, 2018

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To prove for ABC
a=bcos(C)+c cos(B) by using vector law.

Proof

Let us consider that three vectors veca,vecb and vecc are represented respectively in order by three sides BC,CA andAB of a DeltaABC.

This means

vec(BC)=veca

vec(CA)=vecb

vec(AB)=vecc

So we can write

vec(BC)+vec(CA)+vec(AB)=0

=>veca+vecb+vec c=0

=>veca*(veca+vecb+vec c)=0

=>veca*veca+veca*vecb+veca*vec c=0

=>absvecaabsvecacos0^@+absvecaabsvecbcos(pi-C)+absvecaabsvecc cos(pi-B)=0

=>absvecaabsveca*1-absvecaabsvecbcos(C)-absvecaabsvecc cos(B)=0

=>absvecaabsveca=absvecaabsvecbcos(C)+absvecaabsvecc cos(B)

=>absveca=absvecbcos(C)+absvecc cos(B)