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 xxx 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 x and vec y are two non-collinear vectors and ABC is a triangle with side lengths a,b and c satisfying (20a-15b) vec x + (15b-12c) vec y + (12c-20a) vec x xx vec y=0 is:

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 , vec b ,a n d vec c are mutually perpendicular vectors and vec a=alpha( vec axx vec b)+beta( vec bxx vec c)+gamma( vec cxx vec a)a n d[ vec a vec b vec c]=1, then find the value of alpha+beta+gammadot

If vec xa n d vec y are two non-collinear vectors and A B C isa triangle with side lengths a ,b ,a n dc satisfying (20 a-15 b) vec x+(15b-12 c) vec y+(12 c-20 a)( vec xxx vec y)=0, then triangle A B C is a. an acute-angled triangle b. an obtuse-angled triangle c. a right-angled triangle d. an isosceles triangle

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 xxx 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 ,a n d vec c are three non-coplanar non-zero vecrtors, then prove that ( vec a . vec a) vec bxx vec c+( vec a . vec b) vec cxx vec a+( vec a . vec c) vec axx vec b=[ vec b vec c vec a] vec a

If vec a ,a n d 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 ,a n d vec bdot

If vec a , vec b and vec c are non-coplanar vectors, prove that vectors 3vec a-7 vec b-4 vec c ,3 vec a -2 vec b+ vec c and vec a + vec b +2 vec c are coplanar.

If vec a , vec ba n d vec c are non-coplanar vectors, prove that the four points 2 vec a+3 vec b- vec c , vec a-2 vec b+3 vec c ,3 vec a+ 4 vec b-2 vec ca n d vec a-6 vec b+6 vec c are coplanar.

If ( vec axx vec b)^2+( vec a dot vec b)^2=144a n d| vec a|=4, then find the value of | vec b|dot