If `vec a, vec b, vec c, vec d` respectively are the position vectors representing the vertices A, B, C, D of a parallelogram then write `vec d` in terms of `vec a, vec b` and `vec c`

If vectors vec aa n d vec b are two adjacent sides of a parallelogram, then the vector respresenting the altitude of the parallelogram which is the perpendicular to a is vec b+( vec bxx vec a)/(| vec a|^2) b. ( vec a . vec b)/(| vec b|^2) c. vec b-( vec b . vec a)/(| vec a|^2) d. ( vec axx( vec bxx vec a))/(| vec b|^2)

If vec a , vec b , vec c ,a n d vec d are four non-coplanar unit vector such that vec d make equal angles with all the three vectors vec a , vec ba n d vec c , then prove that [ vec d vec a vec b]=[ vec d vec c vec b]=[ vec d vec c vec a]dot

If vec r . vec a= vec r . vec b= vec rdot vec c=1/2 or some nonzero vector vec r , then the area of the triangle whose vertices are A( vec a),B( vec b)a n dC( vec c)i s( vec a , vec b , vec c are non-coplanar ) a. |[ vec a vec b vec c]| b. | vec r| c. |[ vec a vec b vec c] vec r| d. none of these

If ( vec axx vec b)xx( vec bxx vec c)= vec b ,w h e r e vec a , vec b ,a n d vec c are nonzero vectors, then (a) vec a , vec b ,a n d vec c can be coplanar (b) vec a , vec b ,a n d vec c must be coplanar (c) vec a , vec b ,a n d vec c cannot be coplanar (d)none of these

vec a , vec b , vec c are three coplanar unit vectors such that vec a+ vec b+ vec c=0. If three vectors vec p , vec q ,a n d vec r are parallel to vec a , vec b ,a n d vec c , respectively, and have integral but different magnitudes, then among the following options, | vec p+ vec q+ vec r| can take a value equal to a. 1 b. 0 c. sqrt(3) d. 2

If O A B C is a tetrahedron where O is the orogin anf A ,B ,a n dC are the other three vertices with position vectors, vec a , vec b ,a n d vec c respectively, then prove that the centre of the sphere circumscribing the tetrahedron is given by position vector (a^2( vec bxx vec c)+b^2( vec cxx vec a)+c^2( vec axx vec b))/(2[ vec a vec b vec c]) .

If vec(a),vec(b),vec(c) are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of (vec(a)+vec(b))*(vec(b)xxvec(c))+(vec(b)+vec(c))*(vec(c)xxvec(a))+(vec(c)+vec(a))*(vec(a)xxvec(b))

Statement 1: vec a , vec b ,a n d vec c are three mutually perpendicular unit vectors and vec d is a vector such that vec a , vec b , vec ca n d vec d are non-coplanar. If [ vec d vec b vec c]=[ vec d vec a vec b]=[ vec d vec c vec a]=1,t h e n vec d= vec a+ vec b+ vec c Statement 2: [ vec d vec b vec c]=[ vec d vec a vec b]=[ vec d vec c vec a] =>vec d is equally inclined to veca,vecb,vecc.

If vec a , vec ba n d vec c are three mutually perpendicular vectors, then the vector which is equally inclined to these vectors is a. vec a+ vec b+ vec c b. vec a/(| vec a|)+ vec b/(| vec b|)+ vec c/(| vec c|) c. vec a/(| vec a|^2)+ vec b/(| vec b|^2)+ vec c/(| vec c|^2) d. | vec a| vec a-| vec b| vec b+| vec c| vec c

If vec a , vec b , vec ca n d vec d are distinct vectors such that vec axx vec c= vec bxx vec da n d vec axx vec b= vec cxx vec d , prove that ( vec a- vec d)dot (vec b- vec c)!=0,