Search any question & find its solution
Question:
Answered & Verified by Expert
If the components of $\overrightarrow{\mathrm{b}}$ along and perpendicular to $\overrightarrow{\mathrm{a}}$ are
$\lambda \vec{a}$ and $\vec{b}-\lambda \vec{a}$ respectively, what is $\lambda$ equal to ?
Options:
$\lambda \vec{a}$ and $\vec{b}-\lambda \vec{a}$ respectively, what is $\lambda$ equal to ?
Solution:
2129 Upvotes
Verified Answer
The correct answer is:
$\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^{2}}$
We know that the components of $\overrightarrow{\mathrm{b}}$ along $\overrightarrow{\mathrm{a}}$ is $\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\} \overrightarrow{\mathrm{a}}$ and perpendicular to $\overrightarrow{\mathrm{a}}$ is $\overrightarrow{\mathrm{b}}-\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\} \overrightarrow{\mathrm{a}}$
As given : $\left\{\begin{array}{l} \left.\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^{2}}\right\} \vec{a}=\lambda \vec{a}\end{array}\right.$
and $\overrightarrow{\mathrm{b}}-\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\} \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}}-\lambda \overrightarrow{\mathrm{a}}$
$\Rightarrow \lambda=\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\}$
As given : $\left\{\begin{array}{l} \left.\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^{2}}\right\} \vec{a}=\lambda \vec{a}\end{array}\right.$
and $\overrightarrow{\mathrm{b}}-\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\} \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}}-\lambda \overrightarrow{\mathrm{a}}$
$\Rightarrow \lambda=\left\{\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}\right\}$
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.