Prove that ,
` ( vec(a) xx vec(b))^(2) + (vec (a).vec(b))^(2) = |veca|^(2) |vec(b)|^(2)`

Prove that (vec(a)+vec(b)).(vec(a)+vec(b)) = |vec(a)|^(2) + |vec(b)|^(2) , if and only if vec(a) , vec(b) are perpendicular , given vec(a) 1= vec (0) , vec (b) != vec (0)

Prove that , (vec(a) - vec(b)) xx (vec(a) + vec(b)) = 2 ( vec(a) xx vec(b))

Prove that , vec(a) xx ( vec (b) + vec(c)) + vec(b) xx (vec(c) + vec(a)) + vec(c)xx(vec (a) + vec(b)) = vec(0)

Prove that : vec(a) . { vec(b) xx (vec(c) + vec(d))} = vec(a) . (vec(b)xx vec(c)) + vec(a) . ( vec(b) xx vec(d))

Prove that : (vec(a) + vec(b)). { (vec(b)+vec(c))xx (vec(c)+vec(a))}= 2 vec(a) . (vec(b)xxvec(c))

If vec (a) = a_(1) hat (i) + a_(2) hat (j) + a_(3) hat (k) , vec(b) = b_(1) hat (i) + b _(2) hat (j) + b_(3) hat (k) and vec (c ) = c_(1) hat (i) + c_(2) hat (j) + c_(3) hat (k) prove that vec (a ) xx ( vec (b) + vec (c ) ) = vec ( a) xx vec ( b) + vec (a) xx vec(c)

Show that the vectors |vec(a)xxvec(b)|^(2)=|(vec(a).vec(a)" "vec(a).vec(b)),(vec(a).vec(b)" "vec(b).vec(b))|

If vec(a), vec(b) " and " vec(c) are unit vectors satisfying abs(vec(a)-vec(b))^(2)+abs(vec(b)-vec(c))^(2)+abs(vec(c)-vec(a))^(2)=9, " then " abs(2vec(a)+5vec(b)+5vec(c)) is -

Let vec(a),vec(b),vec (c ) be the positions vectors of the vertices of a triangle , prove that the area of the triangle is 1/2| vec(a) xx vec(b) + vec(b) xx vec(c)+ vec(c)xx vec(a)|

If vec aa n d vec b are two vectors and angle between them is theta, then | vec axx vec b|^2+( vec adot vec b)^2=| vec a|^2| vec b|^2 | vec axx vec b|=( vec adot vec b),iftheta=pi//4 vec axx vec b=( vec adot vec b) hat n ,(w h e r e hat n is unit vector, ) if theta=pi//4 ( vec axx vec b)dot( vec a+ vec b)=0