Tamil Nadu Board Solutions for Chapter: Vector Algebra, Exercise 1: EXERCISE

Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side.

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ represent a side and a diagonal of a parallelogram, find the other sides and the other diagonal.

If $\overrightarrow{PO}+\overrightarrow{OQ}=\overrightarrow{QO}+\overrightarrow{OR}$, prove that the points $P,Q,R$ are collinear.

If $D$ is the midpoint of the side $BC$ of a triangle $ABC$, prove that $\overrightarrow{AB}+\overrightarrow{AC}=2\overrightarrow{AD}$.

If $G$ is the centroid of a triangle $ABC$, prove that $\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\overrightarrow{0}$.

Let $A,B$ and $C$ be the vertices of a triangle. Let $D,E$ and $F$ be the midpoints of the sides $BC, CA$ and $AB$ respectively. Show that $\overrightarrow{AD} + \overrightarrow{BE} +\overrightarrow{CF} = 0.$

If $ABCD$ is a quadrilateral and $E$ and $F$ are the midpoints of $AC$ and $BD$ respectively, then prove that $\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{CB}+\overrightarrow{CD}=4\overrightarrow{EF}$.