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  • Provide solution for RD Sharma maths class 12 chapter 22 Algebra of vectors exercise 22.8 question 6 sub question 1 maths textbook solution

Provide solution for RD Sharma maths class 12 chapter 22 Algebra of vectors exercise 22.8 question 6 sub question 1 maths textbook solution

Answers (1)

Answer: vectors are not co-planar

Hint: If in given vectors we can’t express one vector in linear combination of other two, then they are non-coplanar

\RightarrowGiven,

\begin{aligned} &P=3 \hat{i}+\hat{j}-\hat{k}\\ &Q=2 \hat{i}-\hat{j}+7 \hat{k}\\ &R=7 \hat{i}-\hat{j}+23 \hat{k}\\ &\text { then, } \end{aligned}

P\neq xQ+yR then they are non-coplanar

\begin{aligned} &\text { Taking } \\ &P=x Q+y R \\ &3 \hat{i}+\hat{j}-\hat{k}=x(2 \hat{i}-\hat{j}+7 \hat{k})+y(7 \hat{i}-\hat{j}+23 \hat{k}) \\ &3 \hat{i}+\hat{j}-\hat{k}=\hat{i}(2 x+7 y)+\hat{j}(-x-y)+\hat{k}(7 x+23 y) \end{aligned}

Comparing coefficient of \hat{i},\hat{j} & \hat{k}  on both sides

2x+7y=3                                                           .......(1)

-x-y=1                                                            .........(2)

7x+23y=-1                                                    ..........(3)

 To eliminate x from (1) and (2) multiply equation (1) with (-1) and equation (2) with (2) and subtract

-2x-7y=-3

\underline{-2x-2y=2}

         -5y=-5

y=1

Put value of y = 1 in equation (2)

\begin{aligned} &-x-1=1 \\ &\begin{array}{l} -x=2 \\ x=-2 \end{array} \\ &\text { Put } x=-2 \& y=\operatorname{lin}(3) \\ &7 x+23 y=-1 \\ &=7(-2)+23 \\ &=-14+23 \\ &=9 \\ &\neq R \cdot H \cdot S \end{aligned}  

As values of x and y do not satisf the third equation

P\neq xQ+yR

Thus, they are non-coplanar.

 

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