Between the following two statements: Statement I : Let a=i+2j-3k and b=2i+j-k.

Question

Between the following two statements:

Statement I : Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying $\vec{a}	imes \vec{r}=\vec{a}	imes \vec{b}$ and $\vec{a}\cdot \vec{r}=0$ is of magnitude $\sqrt{10}$.

Statement II : In a triangle $ABC,\cos 2A+\cos 2B+\cos 2C\ge -\frac{3}{2}$.

Accepted Answer

$\begin{gathered} \Rightarrow 35=14|\vec{r}{|}^{2} \ \Rightarrow |\vec{r}|=\sqrt{\frac{35}{14}}=\sqrt{\frac{5}{2}}
e \sqrt{10} \end{gathered}$

$	herefore $ Statement I is incorrect

Statement II is correct

$\cos 2A+\cos 2B+\cos 2C\ge -\frac{3}{2}$

Proof: $\because (\overset{	o \leftarrow }{OA}+\overset{	o \leftarrow }{OB}+\overset{	o \leftarrow }{OC})\ge 0\,$

and $|\overset{	o \leftarrow }{OA}{|}^{2}=|\overset{	o \leftarrow }{OB}{|}^{2}=|\overset{	o \leftarrow }{OC}{|}^{2}={R}^{2}\,$

Now, using (1), we get

$|\overset{	o \leftarrow }{OA}{|}^{2}+|\overset{	o \leftarrow }{OB}{|}^{2}+|\overset{	o \leftarrow }{OC}{|}^{2}+2(\overset{	o \leftarrow }{OA}\cdot \overset{	o \leftarrow }{OB}+\overset{	o \leftarrow }{OB}\cdot \overset{	o \leftarrow }{OC}+\overset{	o \leftarrow }{OC}\cdot \overset{	o \leftarrow }{OA})\ge 0$

$\begin{gathered} \Rightarrow 3{R}^{2}+2{R}^{2}(\cos 2A+\cos 2B+\cos 2C)\ge 0 \ \Rightarrow \cos 2A+\cos 2B+\cos 2C\ge -\frac{3}{2} \end{gathered}$

Solutions · 1