" 8.) If "bar(u),bar(v),bar(w)" are three non coplanar vectors "

If bar(u),bar(v),bar(w) are three non coplanar vectors then (bar(u)+bar(v)-bar(w))*{(bar(u)-bar(v))xx(bar(v)-bar(w))}=

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)=

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(c) are three non-coplanar vectors, then the vector equation bar(r)=(1-p-q)bar(a)+pbar(b)+qbar(c) represents

i) If bar(a), bar(b), bar(c) are non coplanar vectors, then prove that the vectors 5bar(a)-6bar(b)+7bar(c), 7bar(a)-8bar(b)+9bar(c) and bar(a)-3bar(b)+5bar(c) are coplanar.

If bar(a),bar(b),bar(c) are non-zero, non -coplanar vectors, then show that the vectors 2bar(a)-5bar(b)+2bar(c),bar(a)+5bar(b)-6bar(c)and3bar(a)-4bar(c) are coplanar.

If bar(a),bar(b),bar(c) and bar(p),bar(q)bar(r) are two sets of three non- coplanar vectors such that bar(a)*bar(p)+bar(b).bar(q)+bar(c).bar(r)=3 then bar(P)=.......bar(q)=......,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 .

let bar(a),bar(b)&bar(c) be three non-zero non coplanar vectors and bar(p),bar(q)&bar(r) be three vectors defined as bar(p)=bar(a)+bar(b)-2bar(c);bar(q)=3bar(a)-2bar(b)+bar(c)&bar(r)=bar(a)-4bar(b)+2bar(c) .If v_(1),v_(2) are the volumes of parallelopiped determined by the vectors bar(a),bar(b),bar(c) and bar(p),bar(q),bar(r) respectively then v_(2):v_(1) is

Let bar(a),bar(b),bar(c) be three non-coplanar vectors and bar(d) be a non-zero vector,which is perpendicularto bar(a)+bar(b)+bar(c). Now,if bar(d)=(sin x)(bar(a)xxbar(b))+(cos y)(bar(b)xxbar(c))+2(bar(c)xxbar(a)) then minimum value of x^(2)+y^(2) is equal to