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^4+b^4+c^4> a b c(a+b+c),w h e r ea ,b ,c > 0.

If a , b , c are real numbers, prove that |[a,b,c],[b,c,a],[c,a,b]|=-(a+b+c)(c+b w+c w^2)(a+b w^2+c w), where w is a complex cube root of unity.

Without expanding, prove that |(a+b x, c+dx, p+q x), (a x+b, c x+d, p x+q),( u, v, w)|=(1-x^2)|(a, c, p),( b, d, q),( u, v, w)| .

If vec u , vec v and vec w are three non-coplanar vectors, then prove that ( vec u+ vec v- vec w) . [ [( vec u- vec v)xx( vec v- vec w)]]= vec u . vec v xx vec w

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-cop0lanar vectors, then prove that ( vec u+ vec v- vec w)dot [( vec u- vec v)xx( vec v- vec w)] = vec u dot (vec v xx vec w)

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

vec u , vec v and vec w are three non-coplanar unit vecrtors and alpha,beta and gamma are the angles between vec u and vec v , vec v and vec w ,a n d vec w and vec u , respectively, and vec x , vec y and vec z are unit vectors along the bisectors of the angles alpha,betaa n dgamma , respectively. Prove that [ vecx xx vec y vec yxx vec z vec zxx vec x]=1/(16)[ vec u vec v vec w]^2sec^2(alpha/2)sec^2(beta/2)sec^2(gamma/2) .

Given that f satisfies |f(u)-f(v)|lt=|u-v| for u and v in [a , b]dot Then |int_a^bf(x)dx-(b-a)f(a)|lt= (a) ((b-a))/2 (b) ((b-a)^2)/2 (b-a)^2 (d) none of these