Search any question & find its solution
Question:
Answered & Verified by Expert
If $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ are unit vectors and $|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|=1$, then $|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|$ is equal to
Options:
Solution:
2835 Upvotes
Verified Answer
The correct answer is:
$\sqrt{3}$
Given, $\quad|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|=1$
$\therefore \quad|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|^{2}=1^{2}$
$\Rightarrow|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}+2|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|=1$
$\Rightarrow \quad 2|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|=1-(1+1)$
$\Rightarrow \quad 2|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|=-1$
Now, $|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|^{2}=|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}-2|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|$ $=1^{2}+1^{2}-(-1)=3$ [from Eq. (i)] $\Rightarrow \quad|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|=\sqrt{3}$
$\therefore \quad|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|^{2}=1^{2}$
$\Rightarrow|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}+2|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|=1$
$\Rightarrow \quad 2|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|=1-(1+1)$
$\Rightarrow \quad 2|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|=-1$
Now, $|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|^{2}=|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}-2|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|$ $=1^{2}+1^{2}-(-1)=3$ [from Eq. (i)] $\Rightarrow \quad|\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}|=\sqrt{3}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.