If S is the circumcentre,G the centroid,and O | vedclass

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(C) Let $P$ be any point in the plane of the triangle $ABC$. The position vector of the centroid $G$ is given by $\overrightarrow {OG} = \frac{1}{3}(\

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$\overrightarrow {SO} $

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(C) Let $P$ be any point in the plane of the triangle $ABC$. The position vector of the centroid $G$ is given by $\overrightarrow {OG} = \frac{1}{3}(\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC})$ if $O$ is the origin. More generally,for any point $P$,$\overrightarrow {PA} + \overrightarrow {PB} + \overrightarrow {PC} = 3\overrightarrow {PG}$. Setting $P = S$ (the circumcentre),we have $\overrightarrow {SA} + \overrightarrow {SB} + \overrightarrow {SC} = 3\overrightarrow {SG}$. Using the Euler line property,the orthocentre $O$,centroid $G$,and circumcentre $S$ are collinear such that $G$ divides $SO$ in the ratio $2:1$,i.e.,$\overrightarrow {SG} = \frac{1}{3}\overrightarrow {SO}$. Substituting this into the equation: $\overrightarrow {SA} + \overrightarrow {SB} + \overrightarrow {SC} = 3(\frac{1}{3}\overrightarrow {SO}) = \overrightarrow {SO}$.

Column-$I$	Column-$II$
$(A)$ In $R^2$,if the magnitude of the projection vector of the vector $\alpha \hat{i}+\beta \hat{j}$ on $\sqrt{3} \hat{i}+\hat{j}$ is $\sqrt{3}$ and if $\alpha=2+\sqrt{3} \beta$,then possible value$(s)$ of $\|\alpha\|$ is (are)	$(P)$ $1$
$(B)$ Let $a$ and $b$ be real numbers such that the function $f(x)=\begin{cases} -3ax^2-2, & x < 1 \\ bx+a^2, & x \geq 1 \end{cases}$ is differentiable for all $x \in R$. Then possible value$(s)$ of $a$ is (are)	$(Q)$ $2$
$(C)$ Let $\omega \neq 1$ be a complex cube root of unity. If $(3-3\omega+2\omega^2)^{4n+3} + (2+3\omega-3\omega^2)^{4n+3} + (-3+2\omega+3\omega^2)^{4n+3}=0$,then possible value$(s)$ of $n$ is (are)	$(R)$ $3$
$(D)$ Let the harmonic mean of two positive real numbers $a$ and $b$ be $4$. If $q$ is a positive real number such that $a, 5, q, b$ is an arithmetic progression,then the value$(s)$ of $\|q-a\|$ is (are)	$(S)$ $4$
	$(T)$ $5$

If $S$ is the circumcentre,$G$ the centroid,and $O$ the orthocentre of a triangle $ABC$,then $\overrightarrow {SA} + \overrightarrow {SB} + \overrightarrow {SC} = $

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