A) \[5\sqrt{2}\]
B) 10
C) 15
D) 5
E) 25
Correct Answer: E
Solution :
\[\overrightarrow{p}\bot (\overrightarrow{q}+\overrightarrow{r})\Rightarrow \overrightarrow{p}.(\overrightarrow{q}+\overrightarrow{r})=0\] \[\Rightarrow \] \[\overrightarrow{p}.\overrightarrow{q}+\overrightarrow{p}.\overrightarrow{r}=0\] ?.. (i) \[\overrightarrow{q}\bot (\overrightarrow{r}+\overrightarrow{p})\Rightarrow \overrightarrow{q}.(\overrightarrow{r}.\overrightarrow{p})=0\] \[\Rightarrow \] \[\overrightarrow{q}.\overrightarrow{r}+\overrightarrow{q}.\overrightarrow{p}=0\] ?. (ii) \[\overrightarrow{r}\bot (\overrightarrow{p}+\overrightarrow{q})\Rightarrow \overrightarrow{r}.(\overrightarrow{p}+\overrightarrow{q})=0\] \[\Rightarrow \] \[\overrightarrow{r}.\overrightarrow{p}+\overrightarrow{r}.\overrightarrow{q}=0\] .. (iii) Adding Eqs. (i), (ii) and (iii), we get \[\overrightarrow{p}.\overrightarrow{q}+\overrightarrow{q}.\overrightarrow{r}+\overrightarrow{r}.\overrightarrow{p}=0\] ...(iv) Now,\[|\overrightarrow{p}+\overrightarrow{q}|=6\Rightarrow (\overrightarrow{p}+\overrightarrow{q}).(\overrightarrow{p}+\overrightarrow{q})=36\] \[\Rightarrow \] \[|\overrightarrow{p}{{|}^{2}}+\overrightarrow{p}\,.\,\overrightarrow{q}+\overrightarrow{q}\,.\,\overrightarrow{p}+|\overrightarrow{q}{{|}^{2}}=36\] ...(v) Similarly, \[|\overrightarrow{q}+\overrightarrow{r}=4\sqrt{3}\] \[\Rightarrow \] \[|\overrightarrow{q}{{|}^{2}}+\overrightarrow{q}.\,\overrightarrow{r}+\overrightarrow{r}.\,\overrightarrow{q}+|\overrightarrow{r}{{|}^{2}}=48\] ...(vi) and \[|\overrightarrow{r}+\overrightarrow{p}|=4\] \[\Rightarrow \] \[|\overrightarrow{r}{{|}^{2}}+\overrightarrow{r}.\overrightarrow{p}+\overrightarrow{p}.\overrightarrow{r}+|\overrightarrow{p}{{|}^{2}}=16\] ..(vii) Adding Eqs. (v), (vi) and (vii), we get \[2|\overrightarrow{p}{{|}^{2}}+2|\overrightarrow{q}{{|}^{2}}+2|\overrightarrow{r}{{|}^{2}}+2(\overrightarrow{p}.\overrightarrow{q}+\overrightarrow{q}.\overrightarrow{r}+\overrightarrow{r}.\overrightarrow{p})\] \[=100\] \[\Rightarrow \] \[|\overrightarrow{p}{{|}^{2}}+|\overrightarrow{q}{{|}^{2}}+|\overrightarrow{r}{{|}^{2}}=\frac{100}{2}=50\] ...(viii) [using Eq.(iv)] Now, \[{{(P+q+r)}^{2}}\] \[=|\overrightarrow{p}|+|\overrightarrow{q}{{|}^{2}}+|\overrightarrow{r}{{|}^{2}}+2(\overrightarrow{p}.\overrightarrow{q}+\overrightarrow{q}.\overrightarrow{r}+\overrightarrow{r}.\overrightarrow{p})\] \[=50\] [using Eqs. (iv) and (viii)] \[\Rightarrow \] \[|\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r}|=5\sqrt{2}\]
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