Question

If A=i+ 2j-3k and B=3i+j+2k ,show that A+B is perpendicular toA-B

Answer 1

GIVEN :–

• Two vectors –

$\\ \bf \to \: \: \overrightarrow{A}= \hat{i}+ 2 \hat{j}-3 \hat{k}$

$\\ \bf \to \: \: \overrightarrow{B}= 3\hat{i}+\hat{j}+2\hat{k}$

TO PROVE :–

$\\ \longrightarrow \: \bf (\overrightarrow{A} + \overrightarrow{B}) \: \: is \: \: perpendicular\: \: to(\overrightarrow{A} - \overrightarrow{B})$

SOLUTION :–

• Two vectors are perpendicular if their vector product is zero.

• Add both vectors –

$\\ \implies \bf \overrightarrow{A} + \overrightarrow{B} =( \hat{i}+ 2 \hat{j}-3 \hat{k})+(3\hat{i}+\hat{j}+2\hat{k})$

$\\ \implies \bf \overrightarrow{A} + \overrightarrow{B} =( \hat{i}+3\hat{i}) + ( 2 \hat{j}+\hat{j}) + (-3 \hat{k}+2\hat{k})$

$\\ \implies \bf \overrightarrow{A} + \overrightarrow{B} =4\hat{i}+ 3\hat{j} - \hat{k}$

• Now substract both vectors –

$\\ \implies \bf \overrightarrow{A} - \overrightarrow{B} =( \hat{i}+ 2 \hat{j}-3 \hat{k}) - (3\hat{i}+\hat{j}+2\hat{k})$

$\\ \implies \bf \overrightarrow{A} - \overrightarrow{B} =( \hat{i} - 3\hat{i}) + ( 2 \hat{j} - \hat{j}) + (-3 \hat{k} - 2\hat{k})$

$\\ \implies \bf \overrightarrow{A} - \overrightarrow{B} = - 2\hat{i}+ \hat{j} - 5\hat{k}$

• Now –

$\\ \implies \bf(\overrightarrow{A} + \overrightarrow{B}).( \overrightarrow{A} - \overrightarrow{B})=(4\hat{i}+ 3\hat{j} - \hat{k}).(- 2\hat{i}+ \hat{j} - 5\hat{k})$

$\\ \implies \bf(\overrightarrow{A} + \overrightarrow{B}).( \overrightarrow{A} - \overrightarrow{B})=(4)( - 2) + (3)(1) + ( - 1)( - 5)$

$\\ \implies \bf(\overrightarrow{A} + \overrightarrow{B}).( \overrightarrow{A} - \overrightarrow{B})= - 8+3+5$

$\\ \implies \bf(\overrightarrow{A} + \overrightarrow{B}).( \overrightarrow{A} - \overrightarrow{B})= - 8+8$

$\\ \large \implies{ \boxed{ \bf(\overrightarrow{A} + \overrightarrow{B}).( \overrightarrow{A} - \overrightarrow{B})=0}}$

$\\ \large \longrightarrow { \boxed{ \boxed{ \bf Hence \: \: Proved }}}$