Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Provide solution for RD Sharma Maths Class 12 Chapter 25 Scale Triple Product Exercise Very Short Answer Question, question 4.

Provide solution for RD Sharma Maths Class 12 Chapter 25 Scale Triple Product Exercise Very Short Answer Question, question 4.

Answers (1)

Answer:

a=1, \frac{1}{2}

Hint:

Use scalar triple product formula.

Given:

 

Vectors \vec{\alpha}=\hat{\imath}+2 \hat{\jmath}+\hat{k}, \vec{\beta}=a \hat{\imath}+\hat{\jmath}+2 \hat{k} \text { and } \vec{\gamma}=\hat{\imath}+2 \hat{\jmath}+a \hat{k} are co planar.

Solution:

We have

\left.\begin{array}{l} \vec{\alpha}=\hat{\imath}+2 \hat{\jmath}+\hat{k} \\ \vec{\beta}=a \hat{\imath}+\hat{\jmath}+2 \hat{k} \\ \vec{\gamma}=\hat{\imath}+2 \hat{\jmath}+a \hat{k} \end{array}\right\}                         Eq.(i)

We know if three vectors are co planar then their scalar triple product is zero.

So, A/Q

α,β and γ are co planar then

\begin{aligned} &{\left[\begin{array}{lll} \vec{\alpha} & \vec{\beta} & \vec{\gamma} \end{array}\right]=0} \\ &\Rightarrow \vec{\alpha} \cdot(\vec{\beta} \times \vec{\gamma})=0 \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad[\because[\vec{a} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]=\vec{a} \cdot(\vec{b} \times \vec{c})] \\ &\Rightarrow(\hat{i}+2 \hat{j}+\hat{k}) \cdot\{(a \hat{i}+\hat{j}+2 \hat{k}) \times(\hat{i}+2 \hat{j}+a \hat{k})\}=0 \; \; \; \; \; \quad[\text { Using }(i)] \end{aligned}

\Rightarrow(\hat{i}+2 \hat{j}+\hat{k}) \cdot\left|\begin{array}{lll} \hat{i} & \hat{j} & \hat{k} \\ a & 1 & 2 \\ 1 & 2 & a \end{array}\right|=0

\Rightarrow(\hat{i}+2 \hat{j}+\hat{k}) \cdot\left\{\hat{i}(a-4)-\hat{j}\left(a^{2}-2\right)+\hat{k}(2 a-1)\right\}=0

\begin{aligned} \Rightarrow(a-4)(\hat{\imath} . \hat{\imath})-\left(a^{2}-2\right)(\hat{\imath} . \hat{\jmath})+(2 a-1)(\hat{\imath} . \hat{k})+2(a-4)(\hat{\jmath} \cdot \hat{\imath})-2\left(a^{2}-2\right)(\hat{\jmath} \cdot \hat{\jmath}) \\ +2(2 a-1)(\hat{\jmath} . \hat{k})+(a-4)(\hat{k} \cdot \hat{\imath})-\left(a^{2}-2\right)(\hat{k} \cdot \hat{\jmath})+(2 a-1)(\hat{k} \cdot \hat{k})=0 \end{aligned}

\begin{aligned} &\Rightarrow(a-4) \cdot 1-0+0+0-2\left(a^{2}-2\right) \cdot 1+0+0-0+(2 a-1) \cdot 1=0 \\ &{\left[\begin{array}{l} \because \hat{\imath} . \hat{\imath}=\hat{\jmath} \cdot \hat{\jmath}=\hat{k} \cdot \hat{k}=1, \hat{\imath} . \hat{\jmath}=\hat{\jmath} \cdot \hat{\imath}=0 \\ \hat{\imath} . \hat{k}=\hat{k} \cdot \hat{\imath}=0, \hat{\jmath} \cdot \hat{k}=\hat{k} \cdot \hat{\jmath}=0 \end{array}\right]} \end{aligned}

\begin{aligned} &\Rightarrow a-4-2 a^{2}+4+2 a-1=0 \\ &\Rightarrow-2 a^{2}+3 a-1=0 \\ &\Rightarrow 2 a^{2}-3 a-1=0 \\ &\Rightarrow 2 a^{2}-2 a-a-1=0 \\ &\Rightarrow 2 a(a-1)-1(a-1)=0 \\ &\Rightarrow(a-1)(2 a-1)=0 \\ &\Rightarrow a-1=0 \text { or } 2 a-1=0 \\ &\Rightarrow a=1 \text { or } a=\frac{1}{2} \end{aligned}

 

Posted by

infoexpert24

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

PM Modi's Pariksha Pe Charcha 2025 to be held next month, registrations open till January 14
anam-khan

CBSE Board Exams 2025: Online portal for availing exemptions by CWSN students opens
anam-khan

Board Exams: States agree to equivalence; no question paper ‘jumbling’ from next year, says PARAKH CEO

Latest Question