Question

If $\overrightarrow{A}=3\hat{i}+3\hat{j}$ and $\overrightarrow{B}=\hat{i}-2\hat{j}+3\hat{k}$ find the magnitude of $\overrightarrow{A}+\overrightarrow{B}and\overrightarrow{A}-\overrightarrow{B}.$

• A. $3\sqrt{2},\sqrt{14}$
• B. $5,\sqrt{29}$
• C. $\sqrt{29},5$
• D. $\sqrt{26},\sqrt{38}$

Answer 1

The correct option is D $\sqrt{26},\sqrt{38}$

$\begin{array}{c}\text{Given:}\overrightarrow{A}=3\hat{ı}+3\hat{ȷ}\\ \overrightarrow{B}=\hat{ı}-2\hat{ȷ}+3\hat{k}\\ \text{To find}:\overrightarrow{A}+\overrightarrow{B}\notin \overrightarrow{A}-\overrightarrow{B}\\ \text{(i)}\overrightarrow{A}+\overrightarrow{B}=3\hat{i}+3\hat{ȷ}+\hat{ı}-2\hat{ȷ}+3\hat{k}\\ =4\hat{ı}+\hat{ȷ}+\hat{k}\end{array}$

$\begin{array}{cc}|\overrightarrow{A}+\overrightarrow{B}{|}^2& =y^2+1^2+3^2\\ |\overrightarrow{A}+\overrightarrow{B}|& =\sqrt{16+1+9}\\ & =\sqrt{26}\end{array}$

$\begin{array}{cc}\overrightarrow{A}-\overrightarrow{B}\text{can be written as :}& \\ \overrightarrow{A}+\left(\overrightarrow{B}\right)& =3\hat{ı}+3\hat{ȷ}+(-(\hat{i}-2\hat{ȷ}+3\hat{k}\left)\right)\\ & =3\hat{ı}+3\hat{ȷ}-\hat{ı}+2\hat{ȷ}-3\hat{k}\\ & =2\hat{ı}+5\hat{ȷ}-3\hat{k}\end{array}$

$\begin{array}{cc}|\overrightarrow{A}+\left(\overrightarrow{B}\right){|}^2& =2^2+5^2+(-3{)}^2\\ |\overrightarrow{A}+\left(\overrightarrow{B}\right)|& =\sqrt{4+25+9}\\ & =\sqrt{38}\end{array}$