Search any question & find its solution
Question:
Answered & Verified by Expert
If $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}-\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}$ are
three coplanar vectors and $|\overrightarrow{\mathrm{c}}|=\sqrt{6}$, then which one of the following is correct ?
Options:
three coplanar vectors and $|\overrightarrow{\mathrm{c}}|=\sqrt{6}$, then which one of the following is correct ?
Solution:
2033 Upvotes
Verified Answer
The correct answer is:
$\quad \mathrm{m}=\pm 2$ and $\mathrm{n}=1$
$\quad \overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$
and $\vec{c}=\hat{i}+m \hat{j}+n \hat{k} ;|\vec{c}|=\sqrt{6}$
Given, $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are coplanar.
So, $\left|\begin{array}{ccc}1 & -1 & 1 \\ 2 & 3 & 2 \\ 1 & \mathrm{~m} & \mathrm{n}\end{array}\right|=0$
$\mathrm{c}_{1} \rightarrow \mathrm{c}_{1}+\mathrm{c}_{2} \quad \mathrm{c}_{3} \rightarrow \mathrm{c}_{3}+\mathrm{c}_{2}$
$\Rightarrow\left|\begin{array}{ccc}0 & 1 & 0 \\ 5 & 3 & 5 \\ 1+\mathrm{m} & \mathrm{m} & \mathrm{m}+\mathrm{n}\end{array}\right|=0$
$\Rightarrow 0+1(5 m+5 n)-(5+5 m)=0$
$\Rightarrow 5 \mathrm{~m}+5 \mathrm{n}-5-5 \mathrm{~m}=0 \Rightarrow 5 \mathrm{n}=5 \Rightarrow \mathrm{n}=1$
$|\vec{c}|=6 \Rightarrow \sqrt{1+m^{2}+n^{2}}=\sqrt{6}$
$\Rightarrow 1+\mathrm{m}^{2}+\mathrm{n}^{2}=6 \Rightarrow 2+\mathrm{m}^{2}=6 \Rightarrow \mathrm{m}^{2}=4 \Rightarrow \mathrm{m}=\pm 2$
and $\vec{c}=\hat{i}+m \hat{j}+n \hat{k} ;|\vec{c}|=\sqrt{6}$
Given, $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are coplanar.
So, $\left|\begin{array}{ccc}1 & -1 & 1 \\ 2 & 3 & 2 \\ 1 & \mathrm{~m} & \mathrm{n}\end{array}\right|=0$
$\mathrm{c}_{1} \rightarrow \mathrm{c}_{1}+\mathrm{c}_{2} \quad \mathrm{c}_{3} \rightarrow \mathrm{c}_{3}+\mathrm{c}_{2}$
$\Rightarrow\left|\begin{array}{ccc}0 & 1 & 0 \\ 5 & 3 & 5 \\ 1+\mathrm{m} & \mathrm{m} & \mathrm{m}+\mathrm{n}\end{array}\right|=0$
$\Rightarrow 0+1(5 m+5 n)-(5+5 m)=0$
$\Rightarrow 5 \mathrm{~m}+5 \mathrm{n}-5-5 \mathrm{~m}=0 \Rightarrow 5 \mathrm{n}=5 \Rightarrow \mathrm{n}=1$
$|\vec{c}|=6 \Rightarrow \sqrt{1+m^{2}+n^{2}}=\sqrt{6}$
$\Rightarrow 1+\mathrm{m}^{2}+\mathrm{n}^{2}=6 \Rightarrow 2+\mathrm{m}^{2}=6 \Rightarrow \mathrm{m}^{2}=4 \Rightarrow \mathrm{m}=\pm 2$
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.