If `[bar(a)bar(b)bar(c)]!=0andbar(p)=(bar(b)xxbar(c))/([bar(a)bar(b)bar(c)]),bar(q)=(bar(c)xxbar(a))/([bar(a)bar(b)bar(c)]),bar(r)=(bar(a)xxbar(b))/([bar(a)bar(b)bar(c)])`, then `bar(a)*bar(p)+bar(b)*bar(q)+bar(c)*bar(r)` is equal to

If bar(a),bar(b),bar(c) are three non coplanar vectors bar(p)=((bar(b)xxbar(c)))/([bar(a)bar(b)bar(c)]),bar(q)=(bar(c)xxbar(a))/([bar(a)bar(b)bar(c)]),bar(r)=(bar(a)xxbar(b))/([bar(a)bar(b)bar(c)]) then (2bar(a)+3bar(b)+4bar(c))*bar(p)+(2bar(b)+3bar(c)+4bar(a))*bar(q)+(2bar(c)+3bar(a)+4bar(b))*bar(r)

If bar(a),bar(b),bar(b),bar(c) are three non coplanar vectors bar(p)=(bar(b)xxbar(c))/([bar(a)bar(b)bar(c)]),bar(q)=(bar(c)xxbar(a))/([bar(a)bar(b)bar(c)]),bar(r)=(bar(a)xxbar(b))/([bar(a)bar(b)bar(c)]) then (2bar(a)+3bar(b)+4bar(c))*bar(p)+(2bar(b)+3bar(c)+4bar(a))bar(q)+(2bar(c)+3bar(a)+4bar(b))*bar(r)=

bar(a),bar(b),bar(c) are three non-coplanar vectors. If bar(p)=(bar(b)xxbar(c))/(bar(a)*(bar(b)xxbar(c))),bar(q)=(bar(c)xxbar(a))/(bar(a)*(bar(b)xxbar(c))),bar(r)=(bar(a)xxbar(b))/(bar(a)*(bar(b)xxbar(c))) , show that bar(a)*bar(p)+bar(b)*bar(q)+bar(c)*bar(r)=3 .

If [bar(a)bar(b)bar(c)]=1 then (bar(a)(bar(b)xxbar(c)))/((bar(c)xxbar(a))*bar(b))+(bar(b)*(bar(c)xxbar(a)))/((bar(a)xxbar(b))*bar(c))+(bar(c)(bar(a)xxbar(b)))/((bar(b)xxbar(c))*bar(a))

If [bar(a)+2bar(b)2bar(b)+bar(c)5bar(c)+bar(a)]=k[bar(a)bar(b)bar(c)]

[bar(a)xxbar(b)bar(c)xxbar(d)bar(e)xxbar(f)] is equal to

(bar(a)+2bar(b)-bar(c))*(bar(a)-bar(b))xx(bar(a)-bar(bar(c)))=

bar(a)xx(bar(b)xxbar(c))+bar(b)xx(bar(c)xxbar(a))+bar(c)xx(bar(a)xxbar(b))=

If [bar(b)xxbar(c)*bar(c)xxbar(a)*bar(a)xxbar(b))]=[bar(a)bar(b)bar(c)]^(k) then the k=

(bar(a)xxbar(i))(bar(b)xxbar(i))+(bar(a)xxbar(j))(bar(b)xxbar(j))+(bar(a)xxbar(k))(bar(b)xxbar(k))=