Let `O A C B` be a parallelogram with `O` at the origin and`O C` a diagonal. Let `D` be the midpoint of `O Adot` using vector methods prove that `B Da n dC O` intersect in the same ratio. Determine this ratio.

If O A B C is a tetrahedron where O is the origin and 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]) .

Let A B C D be a parallelogram whose diagonals intersect at P and let O be the origin. Then prove that vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot

Let A B C D be a p[arallelogram whose diagonals intersect at P and let O be the origin. Then prove that vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot

tan(/_B A O)=3, then find the ratio B C : C Adot

O A B C is regular tetrahedron in which D is the circumcentre of O A B and E is the midpoint of edge A Cdot Prove that D E is equal to half the edge of tetrahedron.

Let A B C be a triangle with A B=A Cdot If D is the midpoint of B C ,E is the foot of the perpendicular drawn from D to A C ,a n dF is the midpoint of D E , then prove that A F is perpendicular to B Edot

Let PQRS be a parallelogram whose diagonals PR and QS intersect at O'. If O is the origin, then (vec(OP)+vec(OQ)+vec(OR)+vec(OS)) is equal to -

A variable line through the point P(2,1) meets the axes at a an d b . Find the locus of the centroid of triangle O A B (where O is the origin).

If A B C D is a rhombus whose diagonals cut at the origin O , then proved that vec O A+ vec O B+ vec O C+ vec O D =0

Statement 1:Let A( vec a),B( vec b)a n dC( vec c) be three points such that vec a=2 hat i+ hat k , vec b=3 hat i- hat j+3 hat ka n d vec c=- hat i+7 hat j-5 hat kdot Then O A B C is a tetrahedron. Statement 2: Let A( vec a),B( vec b)a n dC( vec c) be three points such that vectors vec a , vec ba n d vec c are non-coplanar. Then O A B C is a tetrahedron where O is the origin.