Prove that `[a b c][u v w]=|[a*u, b*u, c*u], [a*v, b*v, c*v], [a*w, b*w, c*w]|`

Prove that /_\ |[a+bx, c+dx, p+qx],[-ax+b, cx+d, px+q],[u,v,w]|=(1-x^2) [[a,c,p],[b,d,q],[u,v,w]]

If hat u , hat v , hat w be three non-coplanar unit vectors with angles between hat u& hat v is alpha between hat v& hat w is beta and between hat w& hat u is gamma . If vec a , vec b , vec c are the unit vectors along angle bisectors of alpha,beta,gamma respectively, then prove that [ vec ax vec b vec bx vec c vec cx vec a]=1/(16)[ hat u hat v hat w]^2sec^2(alpha/2)sec^2(beta/2)sec^2(gamma/2)

If vec u , vec v , vec w are noncoplanar vectors and p, q are real numbers, then the equality [3 vec u ,""p vec v , p vec w]-[p vec v ,"" vec w , q vec u]-[2 vec w ,""q vec v , q vec u]=0 holds for (A) exactly one value of (p, q) (B) exactly two values of (p, q) (C) more than two but not all values of (p, q) (D) all values of (p, q)

Let A={a ,\ b ,\ c} , B={u\ v ,\ w} and let f and g be two functions from A to B and from B to A respectively defined as: f={(a ,\ v), (b ,\ u),\ (c ,\ w)} , g={(u ,\ b),\ (v ,\ a),\ (w ,\ c)}dot Show that f and g both are bijections and find fog and gofdot

If vec u,vec v and vec w are three non-copolanar vectors,then prove that (vec u+vec v-vec w)*(vec u-vec v)xx(vec v-vec w)=vec u*vec v*xxvec w

Let vec u , vec va n d vec w be vectors such that vec u+ vec v+ vec w=0. If | vec u|=3,| vec v|=4a n d| vec w|=5, then vec udot vec v+ vec vdot vec w+ vec wdot vec u is 47 b. -25 c. 0 d. 25

Let vec u,vec v and vec w be such that |vec u|=1,|vec v|=2 and |vec w|=3. If the projection of vec v along vec u is equal to that of vec w along vec u and vectors vec v and vec w are perpendicular to each other,then |vec u-vec v+vec w| equals 2 b.sqrt(7)c.sqrt(14)d.14

Let vec u and vec v be unit vectors such that vec u xxvec v+vec u=vec w and vec w xxvec u=vec v. Find the value of [vec uvec vvec w]