L and M are two points with position vectors `2vec(a) - vec(b)` and `vec(a) + 2vec(b)` respectively. Write the position vectors of a point N which divides the line segment LM in the ratio 2 : 1 externally.

Consider two points P and Q with position vectors vec(OP)=3veca-2vecbandvec(OQ)=veca+vecb . Find the position vector of a point R which divides the line joining P and Q in the ratio 2:1 , (i) intermally , and (ii) externally.

Projection vectors vec a on vec b is

If the position vector of a point A is vec a + 2 vec b and vec a divides AB in the ratio 2:3 , then the position vector of B, is

The position vector of the point which divides the join of points 2 vec a - 3 vec b and vec a + vec b in the ratio 3:1 is

If veca and vecb are position vectors of A and B respectively, then the position vector of a point C in AB produced such ahat vec(AC) = vec(3AB) is :

If vec a and vec b are two unit vectors, then the vector (vec a + vec b) xx (vec a xx vec b) is parallel to the vector

Derive the equation of a line in 3D passing through two points A and B with position vectors vec(a) " and " vec(b) respectively both in vector and Cartesian form.