10.0 Linear Combination of Vectors

10.1 Collinearity of Vectors
10.2 Collinearity of points

Linear Combination is a method of combining things (say variables, polynomials, vectors etc.) using scalar multiplication and addition. It is expressed as $${\text{(scalar)(something 1) + (scalar)(something 2) + (scalar)(something 3) + }}...$$ Now, In case of vectors we can write any vector $\overrightarrow v $ in a linear combination of $\overrightarrow {{v_1}} $, $\overrightarrow {{v_2}} $, $\overrightarrow {{v_3}} $,... if there exists scalar ${a_1}$, ${a_2}$, ${a_3}$... such that $$\overrightarrow v = {a_1}\overrightarrow {{v_1}} + {a_2}\overrightarrow {{v_2}} + {a_3}\overrightarrow {{v_3}} + ...$$

Question 17. Let $\overrightarrow a $, $\overrightarrow b $ and $\overrightarrow c $ be the position vectors of the vertices of the triangle, then find the position vector of centroid of a triangle.

Solution: Let us consider a triangle $ABC$ in which $D$, $E$ and $F$ are the mid points of the sides $BC$, $CA$ and $AB$ respectively and $\overrightarrow a $, $\overrightarrow b $ and $\overrightarrow c $ are position vectors of $A$, $B$ and $C$ respectively.

$$\begin{equation} \begin{aligned} \overrightarrow {OD} = \frac{{\overrightarrow b + \overrightarrow c }}{2} \\ \overrightarrow {OE} = \frac{{\overrightarrow c + \overrightarrow a }}{2} \\ \overrightarrow {OF} = \frac{{\overrightarrow a + \overrightarrow b }}{2} \\\end{aligned} \end{equation} $$

Therefore, position vector of the point dividing median $AD$ in the ratio of $2:1$ is given by section formula as $OG$

$$\begin{equation} \begin{aligned} \overrightarrow {OG} = \frac{{1.\overrightarrow a + 2\{ \frac{{(\overrightarrow b + \overrightarrow c )}}{2}\} }}{{1 + 2}} \\ \overrightarrow {OG} = \frac{{\overrightarrow a + \overrightarrow b + \overrightarrow c }}{3} \\\end{aligned} \end{equation} $$

Similarly, we can find the position vector of point dividing the line $BE$ and $CF$ which are equal as $$\frac{{\overrightarrow a + \overrightarrow b + \overrightarrow c }}{3}$$

Note: Students are advised to remember this point that centroid of triangle is given as $$\frac{{\overrightarrow a + \overrightarrow b + \overrightarrow c }}{3}$$ where $\overrightarrow a $, $\overrightarrow b $ and $\overrightarrow c $ be the position vectors of the vertices of the triangle.