Maharashtra Board Solutions for Chapter: Vectors, Exercise 2: Exercise 5.2

Find the position vector of point $R$ which divides the line joining the points $P$ and $Q$ whose position vectors are $2\hat{i}-\hat{j}+3\hat{k}$ and $-5\hat{i}+2\hat{j}-5\hat{k}$ in the ratio $3: 2$ externally.

If the points $A(3,0,p), B(-1,q,3)$ and $C(-3,3,0)$ are collinear, then find the values of $p$ and $q$.

Prove that the line segments joining mid-point of adjacent sides of a quadrilateral form a parallelogram.

$D$ and $E$ divide sides $BC$ and $CA$ of a triangle $ABC$ in the ratio $2: 3$ respectively. Find the position vector of the point of intersection of $AD$ and $BE$ and the ratio in which this point divides $AD$ and $BE$.

Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.

Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half the sum of parallel sides.

If the centroid of a tetrahedron $OABC$ is $(1,2,-1)$ where $A=(a,2,3),B=(1,b,2)$, $C=(2,1,c)$ respectively, find the distance of $P(a,b,c)$ from the origin.

Find the centroid of tetrahedron with vertices $K(5,-7,0),L(1,5,3),M(4,-6,3),N(6,-4,2)$ ?