cdj8498
Answered

Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

Show that: |A⃗ + B⃗ |² - |A⃗ - B⃗ |² = 4 A⃗.B⃗ .​

Sagot :

Answer:

Taking LHS:-

[tex] { | \overrightarrow{A} + \overrightarrow{B} | }^{2} - { | \overrightarrow{A} - \overrightarrow{B} | }^{2}[/tex]

[tex] \boxed{ \mathsf{using \: \overrightarrow{a} . \overrightarrow{a} = { |a| }^{2} }}[/tex]

[tex] \implies { | \overrightarrow{A} + \overrightarrow{B} | }^{2} = (\overrightarrow{A} + \overrightarrow{B})(\overrightarrow{A} + \overrightarrow{B})[/tex]

[tex] \mathsf{using \: distributive \: property \: of \: vector \: multiplication}[/tex]

eqn 1:-

[tex] \implies \mathsf{ \overrightarrow{A} \overrightarrow{A}+\overrightarrow{A} \overrightarrow{B} + \overrightarrow{B}\overrightarrow{A} + \overrightarrow{B}\overrightarrow{B}}[/tex]

same with the other one

[tex] \boxed{ \mathsf{using \: \overrightarrow{a} . \overrightarrow{a} = { |a| }^{2} }}[/tex]

[tex] { | \overrightarrow{A} - \overrightarrow{B} | }^{2} = (\overrightarrow{A} - \overrightarrow{B})(\overrightarrow{A} - \overrightarrow{B})[/tex]

[tex] \mathsf{using \: distributive \: property \: of \: vector \: multiplication}[/tex]

eqn. 2:-

[tex] \implies \mathsf{ \overrightarrow{A} \overrightarrow{A} - \overrightarrow{A} \overrightarrow{B} - \overrightarrow{B}\overrightarrow{A} + \overrightarrow{B}\overrightarrow{B}}[/tex]

eqn. 1 - eqn. 2 :-

[tex] \implies \mathsf{ \overrightarrow{A} \overrightarrow{A}+\overrightarrow{A} \overrightarrow{B} + \overrightarrow{B}\overrightarrow{A} + \overrightarrow{B}\overrightarrow{B} - ( \overrightarrow{A} \overrightarrow{A} - \overrightarrow{A} \overrightarrow{B} - \overrightarrow{B}\overrightarrow{A} + \overrightarrow{B}\overrightarrow{B})}[/tex]

after distributing the minus sign inside the braces AA and BB get canceled, while the other two add up.

[tex] \implies \mathsf{2 \overrightarrow{A}\overrightarrow{B} + 2\overrightarrow{B}\overrightarrow{A}}[/tex]

[tex] \boxed{ \mathsf{\overrightarrow{A} \overrightarrow{B}= \overrightarrow{B}\overrightarrow{A}} }[/tex]

[tex] \implies \mathsf{ 4\overrightarrow{A}\overrightarrow{B} }[/tex]

LHS = RHS

Hence, Proved! =D

Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.