In isosceles triangles `A B C ,| vec A B|=| vec B C|=8,` a point `E` divides `A B` internally in the ratio 1:3, then find the angle between ` vec C Ea n d vec C A(w h e r e| vec C A|=12)dot`

If | vec a|+| vec b|=| vec c|a n d vec a+ vec b= vec c , then find the angle between vec aa n d vec bdot

If three unit vectors vec a , vec b ,a n d vec c satisfy vec a+ vec b+ vec c=0, then find the angle between vec aa n d vec bdot

If [ vec a vec b vec c]=2, then find the value of [( vec a+2 vec b- vec c)( vec a- vec b)( vec a- vec b- vec c)]dot

The non-zero vectors vec a , vec b and vec c are related by vec a=8 vec b and vec c = −7 vec b , then the angle between vec a and vec c is

If vec a, vec b, vec c are unit vectors such that vec a+ vec b+ vec c =0 , find the value of vec a.vec b+ vec b .vec c + vec c. vec a .

If |(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y)^2(b-z)^2(c-x)^2(c-y)^2(c-a)^2|=0 and vectors vec A , vec B ,a n d vec C , w h e r e vec A=a^2 hat i+a hat j+ hat k , etc, are non-coplanar, then prove that vectors vec X , vec Ya n d vec Z ,w h e r e vec X=x^2 hat i+x hat j+ hat k , etc. may be coplanar.

Let vec a , vec b , vec c be three unit vectors and vec a . vec b= vec a . vec c=0. If the angel between vec b and vec c is pi/3 , then find the value of |[ vec a vec b vec c]| .

Let vec a , vec b ,a n d vec c be any three vectors, then prove that [ vec axx vec b vec bxx vec c vec cxx vec a]=[ vec a vec b vec c]^2dot

If vec a , vec b ,a n d vec c are such that [ vec a vec b vec c]=1, vec c=lambda vec axx vec b , angle, between vec aa n d vec b is (2pi)/3,| vec a|=sqrt(2),| vec b|=sqrt(3)a n d| vec c|=1/(sqrt(3)) , then the angel between vec aa n d vec b is a. pi/6 b. pi/4 c. pi/3 d. pi/2

The position vectors of the vertices A ,Ba n dC of a triangle are three unit vectors vec a , vec b ,a n d vec c , respectively. A vector vec d is such that vecd dot vec a= vecd dot vec b= vec d dot vec ca n d vec d=lambda( vec b+ vec c)dot Then triangle A B C is a. acute angled b. obtuse angled c. right angled d. none of these