If ` vec x , vec y` are two non-zero and non-collinear vectors satisfying `[(a-2)alpha^2+(b-3)alpha+c] vec x+[(a-2)beta^2+(b-3)beta+c] vec y+[(a-2)gamma^2+(b-3)gamma+c]( vec x xx vec y)=0, w h e r ealpha,beta,gamma` are three distinct real numbers, then find the value of `(a^2+b^2+c^2-4)dot`

If vec a , vec ba n d vec c are unit vectors satisfying | vec a- vec b|^2+| vec b- vec c|^2+| vec c- vec a|^2=9, then |2 vec a+5 vec b+5 vec c| is.

If vec a and vec b be two non-collinear unit vector such that vec axx( vec axx vec b)=1/2 vec b , then find the angle between vec a and vec b .

Let vec aa n d vec b be two non-zero perpendicular vectors. A vecrtor vec x satisfying the equation vec x xx vec b= vec a is vecx= beta vecb-1/|b|^2 vecaxx vecb then beta can be

If vec xa n d vec y are two non-collinear vectors and a, b, and c represent the sides of a A B C satisfying (a-b) vec x+(b-c) vec y+(c-a)( vec x xx vec y)=0, then A B C is (where vec xxx vec y is perpendicular to the plane of xa n dy ) a. an acute-angled triangle b. an obtuse-angled triangle c. a right-angled triangle d. a scalene triangle

If vec a , vec b and vec c are any three non-coplanar vectors, then prove that points are collinear: veca+ vecb+vecc , 4veca+3vecb ,10veca+7vec b-2vecc .

If alpha=2vec(i) +3vec(j) -5vec(k) and beta= vec(i)-vec(j) , then find the value of alpha . beta

If vec a , vec b , vec c are mutually perpendicular unit vectors, find |2 vec a+ vec b+ vec c|dot

If vec a, vec b, vec c are unit vectors satisfying the condition veca+ vec b+ vec c= 0 then show that vec a. vec b+ vec b.vec c+ vec c. vec a= -3//2 .

If vec(alpha) and vec (beta) are perpendicular to each other , show that | vec(alpha) + vec(beta)|^(2) = |vec(alpha)-vec(beta)|^(2)