Question

Q:- If $\overrightarrow{P}=3\hat{i}-4\hat{j}$ and $\overrightarrow{Q}=8\hat{i}+86\hat{j}$. Find the magnitude and direction of $\overrightarrow{P}+\overrightarrow{Q}$

Answer 1

Given that,

Vectors

$\begin{align}

$P=3\hat{i}-4\hat{j}$

$Q=8\hat{i}+86\hat{j}$

Now, the magnitude of the vector of $\overrightarrow{P}+\overrightarrow{Q}$

$\overrightarrow{P}+\overrightarrow{Q}=(3i-4j)+(8i+6j)$

$\overrightarrow{P}+\overrightarrow{Q}=11\hat{i}+2\hat{j}$

$|\overrightarrow{P}+\overrightarrow{Q}|=\sqrt{{\left(11\right)}^2+{\left(2\right)}^2}$

$|\overrightarrow{P}+\overrightarrow{Q}|=\sqrt{125}$

$|\overrightarrow{P}+\overrightarrow{Q}|=11.18$

Now, the direction is

$\theta ={\tan }^{-1}\frac{|Q|}{|P|}$

$\theta ={\tan }^{-1}\frac{10}{5}$

$\theta ={\tan }^{-1}2$

$\theta ={63.4}^0$

Hence, the magnitude of the vector is $11.18$ and the direction about ${63.4}^0$