Question

$\overrightarrow{A}$ = 3 $\hat{i}$ + 4 $\hat{j}$ and $\overrightarrow{B}$ = 5 $\hat{i}$ + 2 $\hat{j}$. What is unit vector along the direction of $\overrightarrow{A}$ + $\overrightarrow{B}$ ?

• A. $\frac{4}{5}$ $\hat{i}$ + $\frac{3}{5}$ $\hat{j}$
• B. $\frac{3}{5}$ $\hat{i}$ + $\frac{4}{5}$ $\hat{j}$
• C. $\frac{5}{4}$ $\hat{i}$ + $\frac{3}{5}$ $\hat{j}$
• D. $\frac{4}{5}$ $\hat{i}$ + $\frac{5}{3}$ $\hat{j}$

Answer 1

The correct option is A $\frac{4}{5}$ $\hat{i}$ + $\frac{3}{5}$ $\hat{j}$

Given:
$\overrightarrow{A}$ = 3 $\hat{i}$ + 4 $\hat{j}$
$\overrightarrow{B}$ = 5 $\hat{i}$ + 2 $\hat{j}$
Then, $\overrightarrow{A}$ + $\overrightarrow{B}$
= (3 $\hat{i}$ + 4 $\hat{j}$) + (5 $\hat{i}$ + 2 $\hat{j}$)
= 8 $\hat{i}$ + 6 $\hat{j}$

unit vector along $\overrightarrow{A}$ + $\overrightarrow{B}$
= $\frac{\overrightarrow{A}+\overrightarrow{B}}{|\overrightarrow{A}+\overrightarrow{B}|}$
= $\frac{8\hat{i}+6\hat{j}}{\sqrt{(8{)}^2+(6{)}^2}}$
= $\frac{8}{10}$ $\hat{i}$ + $\frac{6}{10}$ $\hat{j}$
= $\frac{4}{5}$ $\hat{i}$ + $\frac{3}{5}$ $\hat{j}$