The vectors i+j+2 k, i+ j-k and 2 i - j + k are coplanar if - Vidyadip

MCQ

The vectors $\lambda \hat{i}+\hat{j}+2 \hat{k}, \hat{i}+\lambda \hat{j}-\hat{k}$ and $2 \hat{ i }-\hat{ j }+\lambda \hat{ k }$ are coplanar if

✓
$\lambda=-2$
B
$\lambda=0$
C
$\lambda=2$
D
$\lambda=1$

✓

Answer

Correct option: A.

$\lambda=-2$

(A) Let $\overline{ a }, \overline{ b }$ and $\overline{ c }$ be the given vectors.
The given vectors are coplanar.
$\therefore\left|\begin{array}{ccc}\lambda & 1 & 2 \\ 1 & \lambda & -1 \\ 2 & -1 & \lambda\end{array}\right|=0$
$\begin{array}{l}\Rightarrow \lambda\left(\lambda^2-1\right)-(\lambda+2)+2(-1-2 \lambda)=0 \\ \Rightarrow \lambda^3-6 \lambda-4=0 \\ \Rightarrow(\lambda+2)\left(\lambda^2-2 \lambda-2\right)=0\end{array}$
$\Rightarrow \lambda=-2$ or $\lambda=\frac{2 \pm \sqrt{4+8}}{2}=1 \pm \sqrt{3}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Vectors→Vectors — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $5 \cos 2 \theta+2 \cos ^2 \frac{\theta}{2}+1=0,-\pi<\theta<\pi$, then $\theta=$

If the probability density function of a continuous random variable X is
$\begin{aligned}\mathrm{f}(x) & =3\left(1-2 x^2\right) ; & & 0<x<1 \\& =0 ; & & \text { otherwise }\end{aligned}$
Then $\mathrm{P}\left(\frac{1}{4}<\mathrm{X}<\frac{1}{3}\right)=$

Let $\vec{\text{a}}=\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},\vec{\text{b}}=\text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}$ be three zero vectors such that $\vec{\text{c}}$ is a unit vector perpendicular to both $\vec{\text{a}}$ and $\vec{\text{b}}.$ If the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$ is $\frac{\pi}{6},$ then $\begin{vmatrix}\text{a}_1&\text{a}_2&\text{a}_3\\\text{b}_1&\text{b}_2&\text{b}_3\\\text{c}_1&\text{c}_2&\text{c}_3 \end{vmatrix}^2$ is equal to:

If the direction cosines of a line are $\frac{1}{c}, \frac{1}{c}, \frac{1}{c}$, then

If the planes $x+3 y+k z=0$ and $3 x+y-2 z=0$ are at right angles, then the value of $k$ is

p: There are clouds in the sky and q: it is not raining. The symbolic form is

$\int_0^{2 x}(\sin x+|\sin x|) d x=$

A bag containe $5$ black, $4$ white balls and $3$ red balls. if a ball is selected randomwise, the probability that it is black or red ball is,

If slope of one of the lines $a x^2+2 h x y+ b y^2=0$ is 5 times that of the other, then $5 h^2=$

The solution of $\frac{ d y}{ d x}=1$ is