Let A (3, 0, -1), B (2, 10, 6) and C (1, 2, 1) be the vertices


Let $\mathrm{A}(3,0,-1), \mathrm{B}(2,10,6)$ and $\mathrm{C}(1,2,1)$ be the vertices of a triangle and $\mathrm{M}$ be the midpoint of $\mathrm{AC}$. If $\mathrm{G}$ divides $\mathrm{BM}$ in the ratio, $2: 1$, then

$\cos (\angle \mathrm{GOA})$ (O being the origin) is equal to :

  1. $\frac{1}{\sqrt{30}}$

  2. $\frac{1}{6 \sqrt{10}}$

  3. $\frac{1}{\sqrt{15}}$

  4. $\frac{1}{2 \sqrt{15}}$

Correct Option: , 3


$G$ is the centroid of $\triangle A B C$

$G \equiv(2,4,2)$

$\overrightarrow{\mathrm{OG}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$

$\overrightarrow{\mathrm{OA}}=3 \hat{\mathrm{i}}-\hat{\mathrm{k}}$

$\cos (\angle \mathrm{GOA})=\frac{\overrightarrow{\mathrm{OG}} \cdot \overrightarrow{\mathrm{OA}}}{|\overrightarrow{\mathrm{OG}}||\overrightarrow{\mathrm{OA}}|}=\frac{1}{\sqrt{15}}$

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