Search any question & find its solution
Question:
Answered & Verified by Expert
If $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}+2 \hat{k}$, then find the unit vector in the direction of
(i) $6 \vec{b}$
(ii) $2 \vec{a}-\vec{b}$
(i) $6 \vec{b}$
(ii) $2 \vec{a}-\vec{b}$
Solution:
2423 Upvotes
Verified Answer
Here, $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}+2 \hat{k}$
(i) Given $6 \vec{b}=12 \hat{i}+6 \hat{j}+12 \hat{k}$
$\therefore$ Unit vector in the direction of $6 \vec{b}=\frac{6 \vec{b}}{|6 \vec{b}|}$ $=\frac{12 \hat{i}+6 \hat{j}+12 \hat{k}}{\sqrt{12^2+6^2+12^2}}=\frac{2 \hat{i}+\hat{j}+2 \hat{k}}{3}=\frac{2 \hat{i}+\hat{j}+2 \hat{k}}{3}$
(ii) Since, $2 \vec{a}-\vec{b}=2(\hat{i}+\hat{j}+2 \hat{k})-(2 \hat{i}+\hat{j}+2 \hat{k})$
$$
=2 \hat{i}+2 \hat{j}+4 \hat{k}-2 \hat{i}-\hat{j}-2 \hat{k}=\hat{j}+2 \hat{k}
$$
$\therefore$ Unit vector in the direction of
$$
2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}=\frac{2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}}{|2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|}=\frac{(\hat{\mathrm{j}}+2 \hat{\mathrm{k}})}{\sqrt{0+1+4}}=\frac{(\hat{\mathrm{j}}+2 \hat{\mathrm{k}})}{\sqrt{5}}
$$
(i) Given $6 \vec{b}=12 \hat{i}+6 \hat{j}+12 \hat{k}$
$\therefore$ Unit vector in the direction of $6 \vec{b}=\frac{6 \vec{b}}{|6 \vec{b}|}$ $=\frac{12 \hat{i}+6 \hat{j}+12 \hat{k}}{\sqrt{12^2+6^2+12^2}}=\frac{2 \hat{i}+\hat{j}+2 \hat{k}}{3}=\frac{2 \hat{i}+\hat{j}+2 \hat{k}}{3}$
(ii) Since, $2 \vec{a}-\vec{b}=2(\hat{i}+\hat{j}+2 \hat{k})-(2 \hat{i}+\hat{j}+2 \hat{k})$
$$
=2 \hat{i}+2 \hat{j}+4 \hat{k}-2 \hat{i}-\hat{j}-2 \hat{k}=\hat{j}+2 \hat{k}
$$
$\therefore$ Unit vector in the direction of
$$
2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}=\frac{2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}}{|2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|}=\frac{(\hat{\mathrm{j}}+2 \hat{\mathrm{k}})}{\sqrt{0+1+4}}=\frac{(\hat{\mathrm{j}}+2 \hat{\mathrm{k}})}{\sqrt{5}}
$$
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.