Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Vector Algebra
0 votes

Prove that \( (\overrightarrow a + \overrightarrow b) ⋅ (\overrightarrow a + \overrightarrow b) =| \overrightarrow a |^2 + |\overrightarrow b |^2\), if and only if \(\overrightarrow a,\: \overrightarrow b\) are perpendicular, given \(\overrightarrow a ≠ 0, \overrightarrow b ≠ 0\).

Can you answer this question?

1 Answer

0 votes
  • $\overrightarrow a.\overrightarrow b=|\overrightarrow a||\overrightarrow b|\cos\theta$
  • If $\overrightarrow a\perp \overrightarrow b$ then $\overrightarrow a.\overrightarrow b=0$
Step 1:
Let us first determine $(\overrightarrow a+\overrightarrow b).(\overrightarrow a+\overrightarrow b)$
$(\overrightarrow a+\overrightarrow b).(\overrightarrow a+\overrightarrow b)=\overrightarrow a(\overrightarrow a+\overrightarrow b)\overrightarrow b(\overrightarrow a+\overrightarrow b)$
$\qquad\qquad\qquad\quad=\overrightarrow a.\overrightarrow a+\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow a+\overrightarrow b.\overrightarrow b$
$\qquad\qquad\qquad\quad\;\;=|\overrightarrow a|^2+2\overrightarrow a.\overrightarrow b+|\overrightarrow b|^2$
Step 2:
Let us consider if $\overrightarrow a\perp \overrightarrow b$
If $(\overrightarrow a \perp \overrightarrow b)$ then $\overrightarrow a.\overrightarrow b=0$
Therefore $(\overrightarrow a+\overrightarrow b).(\overrightarrow a+\overrightarrow b)=|\overrightarrow a|^2+|\overrightarrow b|^2$
Step 3:
If $\overrightarrow a$ is not perpendicular to $\overrightarrow b$ then,$\overrightarrow a.\overrightarrow b\neq 0$,Similarly $\overrightarrow b.\overrightarrow a\neq 0$
Then $(\overrightarrow a+\overrightarrow b).(\overrightarrow a.\overrightarrow b)=|\overrightarrow a|^2+|\overrightarrow b|^2+2\overrightarrow a.\overrightarrow b\neq |\overrightarrow a|^2+|\overrightarrow b|^2$
Hence only if $\overrightarrow a\perp \overrightarrow b,(\overrightarrow a+\overrightarrow b)=|\overrightarrow a|^2+|\overrightarrow b|^2$
answered May 24, 2013 by sreemathi.v
edited May 24, 2013 by sreemathi.v

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App