Question

lf the vectors $\overrightarrow{A}=2\hat{i}+4\hat{j}$ and $\overrightarrow{B}=5\hat{i}-P\hat{j}$ are parallel to each other, the magnitude of $\overrightarrow{B}$ is:

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

Answer 1

The correct option is A $5\sqrt{5}$

$\overline{A}.\overline{B}=|\overrightarrow{A}||\overrightarrow{B}|$
$10-4P=2\sqrt{5}\times \sqrt{25+P^2}$

Squaring on both sides:
$100+16P^2-80P=500+20P^2$
$4P^2+80P+400=0$
$P^2+20P+100=0$
$P=-10$

So, magnitude of $\overline{B}=\sqrt{25+P^2}=\sqrt{25+100}=\sqrt{125}=5\sqrt{5}$