Prove that, in a parallelogram `PQRS, vec(PR) - vec(QS) = vec(2PQ)`

Let ABCD be a parallelogram such that vec(AB)= vec(q), vec(AD) = vec(P) and angleBAD be an acute angle. If vec(r) is the vector that coincides with the altitude directed from the vertex B to the side AD, then vec(r) is given by-

Let vec(PR) = 3hat(i) + hat(j) - 2hat(k) and vec(SQ) = hat(i) - 3hat(j) - 4hat(k) determine diagonals of aparallelogram PQRS and vec(PT) = hat(i) + 2hat(j) + 3hat(k) be another vector. Then the volume of the parallelopiped determined by the vectors vec(PT), vec(PQ) and vec(PS) is

bar(AC) " and " bar(BD) are the diagonals of the parallelogram ABCD. Prove that, vec(AC)+vec(BD)=2vec(BC) " and " vec(AC)-vec(BD)=2vec(AB)

Prove that , (vec(a) - vec(b)) xx (vec(a) + vec(b)) = 2 ( vec(a) xx vec(b))

Let vec(a),vec(b),vec (c ) be the positions vectors of the vertices of a triangle , prove that the area of the triangle is 1/2| vec(a) xx vec(b) + vec(b) xx vec(c)+ vec(c)xx vec(a)|

Prove that : vec(a) . { vec(b) xx (vec(c) + vec(d))} = vec(a) . (vec(b)xx vec(c)) + vec(a) . ( vec(b) xx vec(d))

Prove that , vec(a) xx ( vec (b) + vec(c)) + vec(b) xx (vec(c) + vec(a)) + vec(c)xx(vec (a) + vec(b)) = vec(0)

Let DeltaPQR be a triangle. Let vec(a) = vec(QR), vec(b) = vec(RP) and vec(c) = vec(PQ) . If |vec(a)| = 12, |vec(b)| = 4 sqrt3 and vec(b).vec(c) = 24 , then which of the following is (are) true?

Prove that , ( vec(a) xx vec(b))^(2) + (vec (a).vec(b))^(2) = |veca|^(2) |vec(b)|^(2)

Two vectors vec(a) and vec(b) are such that |vec(a)| = 2 , |vec(b)| = 1 and vec(a) . vec(b) = 1 , then find the value of (3 vec(a) - 5 vec(b)).(2 vec(a) + 7vec(b))