Prove that `1/2 vec(AC)xxvec(BD)` represents the vector area of the plane quadrilateral ABCD.

In Fig. 9.22, ABCD is a quadrilateral and BE || AC and also BE meets DC produced at E. Show that area of Delta ADE is equal to the area of the quadrilateral ABCD.

If A(-5, 7), B(- 4, -5), C(-1, -6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD,

P,Q,R and S are respectively the mid points of the sides AB,BC,CD and Da of a quadrilateral ABCD in which AC=BD and ACbotBD . Prove that PQRS is a square.

If vec a , vec b , vec c , vec d are the position vector of point A , B , C and D , respectively referred to the same origin O such that no three of these point are collinear and vec a + vec c = vec b + vec d , than prove that quadrilateral A B C D is a parallelogram.

If the mid-point of the sides of a quadrilateral are joined in order, prove that the area of the parallelogram so formed will be half of the area of the given quadrilateral.

In a quadrilateral ABCD, angleA,angleB,angleC and angleD in the ratio 1:2:2:4. find the measure of each angle of the quadrilateral.

Solve for vec x , the equation: vec BE + vec x + vec ED = vec BD , where ABCD is a quadilateral whose diagonals intersects in E.

If A,B,C and D are any four points in space prove that |quad vec (AB) xx vec (CD)+ vec (BC) xx vec (AD) + vec (CA) xx vec (BD)|=4(area of Delta ABC) .

ABCD is a quadilateral in which [BC] is parallel to [AD] and the ratio of the lengths BC: AD :: 4: 7 . Taking vec AB and vec AD as repressentatives of vectors vec v and 7 vec u respectively, find the vectors represented by vec BC