Question

Let $P(1,2,3)$ and $Q(-1,-2,-3)$ be the two points and let $O$ be the origin. Then, $|PQ+OP|$ is equal to

• A. $\sqrt{13}$
• B. $\sqrt{14}$
• C. $\sqrt{24}$
• D. $\sqrt{12}$
• E. $\sqrt{8}$

Answer 1

Answer: (B)

$\sqrt{14}$

Given that, $P=(1,2,3),Q=(-1,-2,-3)$ and $O=(0,0,0)$.

Then, $PQ=(-1-1)\hat{i}+(-2-2)\hat{j}+(-3-3)\hat{k}$
$=-2\hat{i}-4\hat{j}-6\hat{k}$
and $OP=(1-0)\hat{i}+(2-0)\hat{j}+(3-0)\hat{k}$
$=\hat{i}+2\hat{j}+3\hat{k}$
Now, $PQ+OP=(-2\hat{i}-4\hat{j}-6\hat{k})+(\hat{i}+2\hat{j}+3\hat{k})$
$=-\hat{i}-2\hat{j}-3\hat{k}$
$\Rightarrow |PQ+OP|=\sqrt{{(-1)}^2+{(-2)}^2+{(-3)}^2}=\sqrt{14}$

Alternate Method

$PQ=OQ-OP$
$\Rightarrow PQ+OP=OQ$
Now, $OQ=-\hat{i}-2\hat{j}-3\hat{k}$
$\left|OQ\right|=\sqrt{{(-1)}^2+{(-2)}^2+{(-3)}^2}$ $=\sqrt{14}$
Therefore, $|PQ+OP|=\left|OQ\right|$ $=\sqrt{14}$