Download App

VECTOR ALGEBRA — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVECTOR ALGEBRA3 Marks
Question
Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},\vec{\text{b}}=3\hat{\text{i}}+\lambda\hat{\text{j}}+\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}$
✓

Answer

Given:
$\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$
$\vec{\text{b}}=3\hat{\text{i}}+\lambda\hat{\text{j}}+\hat{\text{k}}$
$\vec{\text{c}}=\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}$
We know that vectors $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are coplanar iff $\Big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\Big]=0.$
It is given that $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are coplanar.
$\therefore\Big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\Big]=0$
$\Rightarrow\begin{vmatrix}1&2&-3\\3&\lambda&1\\1&2&2 \end{vmatrix}=0$
$\Rightarrow1(2\lambda-2)-2(6-1)-3(6-\lambda)=0$
$\Rightarrow5\lambda-30=0$
$\Rightarrow\lambda=6$

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 "3 Marks Question" from VECTOR ALGEBRA→VECTOR ALGEBRA — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

$\overrightarrow{a} = \hat{i} + 2\hat{j} - 3\hat{k}, \overrightarrow{b} = 3\hat{i} - \hat{j} + 2\hat{k}, \text{show that}\bigg(\overrightarrow{a} +\overrightarrow{b}\bigg) \text{and} \bigg(\overrightarrow{a} -\overrightarrow{b}\bigg)$ are perpendicular to each other.
→
If $\text{A}=\begin{bmatrix}2 & 3 \\ 5 & -2 \end{bmatrix}$ be sech that $A^{-1} = kA,$ then find the value of $k.$
→
If A = {1, 2, 3, 4} define relations on A which have properties of being:
Reflexive, transitive but not symmetric.
→
An instructor has a test bank consisting of 300 easy True/False questions, 200 difficult True/False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the test bank, what is the probability that it will be an easy question given that it is a multiple choice question?
→
If $\text{y}=\sin(\log\text{x})$ prove that $\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$
→
Find the vector equation of the line passing through the point A(1, 2, -1) and parallel to the line 5x - 25 = 14 - 7y = 35z.
→
Show that the points whose position vectors are as given below are collinear:
$3\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}},\ \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$
→
Evaluate: $\int \frac{x+2}{\sqrt{x^2+2 x-1}} d x$
→
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\left( {2\vec a + \vec b} \right)$and$\left( {\vec a - 3\vec b} \right)$ externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.
→
Write the vector equation of the line passing through the point (1, -2, -3) and normal to the plane $\vec{\text{r}}.(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})=5.$
→