Search any question & find its solution
Question:
Answered & Verified by Expert
If $|\overrightarrow{\mathbf{a}}|=3,|\overrightarrow{\mathbf{b}}|=4$, then for what value of 1 is $(\overrightarrow{\mathbf{a}}+\lambda \overrightarrow{\mathbf{b}})$
perpendicular to $(\overrightarrow{\mathbf{a}}-\lambda \overrightarrow{\mathbf{b}})$ ?
Options:
perpendicular to $(\overrightarrow{\mathbf{a}}-\lambda \overrightarrow{\mathbf{b}})$ ?
Solution:
2589 Upvotes
Verified Answer
The correct answer is:
$\frac{3}{4}$
$\because(\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}})$ is perpendicular to $(\overrightarrow{\mathrm{a}}-\lambda \overrightarrow{\mathrm{b}})$, their dot
product is zero, so, $(\vec{a}+\lambda \vec{b}) \cdot(\vec{a}-\lambda \vec{b})=0$
$\Rightarrow|\overrightarrow{\mathrm{a}}|^{2}-\lambda^{2}|\overrightarrow{\mathrm{b}}|^{2}-\lambda \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\lambda \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{a}}=0$
$\Rightarrow|\overrightarrow{\mathrm{a}}|^{2}-\lambda^{2}|\overrightarrow{\mathrm{b}}|^{2}=0 \quad(\because \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{b} \cdot \mathrm{a}})$
$\Rightarrow 9-16 \lambda^{2}=0$
$\Rightarrow \lambda=\pm \frac{3}{4} \quad \lambda=\frac{3}{4}$ matches with the given option.
product is zero, so, $(\vec{a}+\lambda \vec{b}) \cdot(\vec{a}-\lambda \vec{b})=0$
$\Rightarrow|\overrightarrow{\mathrm{a}}|^{2}-\lambda^{2}|\overrightarrow{\mathrm{b}}|^{2}-\lambda \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\lambda \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{a}}=0$
$\Rightarrow|\overrightarrow{\mathrm{a}}|^{2}-\lambda^{2}|\overrightarrow{\mathrm{b}}|^{2}=0 \quad(\because \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{b} \cdot \mathrm{a}})$
$\Rightarrow 9-16 \lambda^{2}=0$
$\Rightarrow \lambda=\pm \frac{3}{4} \quad \lambda=\frac{3}{4}$ matches with the given option.
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.