Question

If $\overrightarrow{p},\overrightarrow{q}$ and $\overrightarrow{r}$ are perpendicular to $\overrightarrow{q}+\overrightarrow{r},\overrightarrow{r}+\overrightarrow{p}$ and $\overrightarrow{p}+\overrightarrow{q}$ respectively and if $|\overrightarrow{p}+\overrightarrow{q}|=6,$ $|\overrightarrow{q}+\overrightarrow{r}|=4\sqrt{3}$ and $|\overrightarrow{r}+\overrightarrow{p}|=4,$ then $|\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r}|$ is

• A. $5\sqrt{2}$
• B. $10$
• C. $15$
• D. $5$
• E. $25$

Answer 1

Answer: (E)

$\overrightarrow{p}\perp (\overrightarrow{q}+\overrightarrow{r})\Rightarrow \overrightarrow{p}.(\overrightarrow{q}+\overrightarrow{r})=0$
$\Rightarrow$ $\overrightarrow{p}.\overrightarrow{q}+\overrightarrow{p}.\overrightarrow{r}=0$ .. (i)
$\overrightarrow{q}\perp (\overrightarrow{r}+\overrightarrow{p})\Rightarrow \overrightarrow{q}.(\overrightarrow{r}.\overrightarrow{p})=0$
$\Rightarrow$ $\overrightarrow{q}.\overrightarrow{r}+\overrightarrow{q}.\overrightarrow{p}=0$ .. (ii)
$\overrightarrow{r}\perp (\overrightarrow{p}+\overrightarrow{q})\Rightarrow \overrightarrow{r}.(\overrightarrow{p}+\overrightarrow{q})=0$
$\Rightarrow$ $\overrightarrow{r}.\overrightarrow{p}+\overrightarrow{r}.\overrightarrow{q}=0$ .. (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)]
$\Rightarrow$ $|\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r}|=5\sqrt{2}$