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  • Provide solution for RD Sharma maths class 12 chapter 22 Algebra of vectors exercise 22.8 question 3 sub question 1 maths textbook solution

Provide solution for RD Sharma maths class 12 chapter 22 Algebra of vectors exercise 22.8 question 3 sub question 1 maths textbook solution

Answers (1)

Answer: Vectors are co-planar

Hint: Vectors are coplanar if we express them as a linear combination of other two.

\begin{aligned} &\Rightarrow 5 \vec{a}+6 \vec{b}+7 \vec{c}=x(7 \vec{a}-8 \vec{b}+9 \vec{c})+y(3 \vec{a}+20 \vec{b}+5 \vec{c}) \\ &\Rightarrow 5 \vec{a}+6 \vec{b}+7 \vec{c}=\vec{a}(7 x+3 y)+\vec{b}(-8 x+20 y)+\vec{c}(9 x+5 y) \end{aligned}

Comparing the equations

\begin{aligned} &5=7 x+3 y\\ &7 x+3 y=5\\ \end{aligned}                          .....(1)

-8 x+20 y=6\\                   .......(2)

9 x+5 y=7                        ........(3)  

 Solving equation (1) and (2) by elimination method

\begin{aligned} &7 x+3 y=5\\ \end{aligned}                           ............\left ( 1 \right )\times \left ( -8 \right )

-8 x+20 y=6\\                    .............\left ( 2 \right )\times \left ( 7 \right )

\underline{\begin{gathered} \begin{array}{r} -56 x-24 y=-40 \\ -56 x+140 y=42 \\ \end{array} \end{gathered}}

-164 y=-82 \\ y=\frac{-82}{-164} \\ y=\frac{1}{2}

Put value of y in (1)

\begin{aligned} &7 x+3\left(\frac{1}{2}\right)=5 \\ &7 x+\frac{3}{2}=5 \\ &7 x=5-\frac{3}{2} \\ &7 x=\frac{10-3}{2} \\ &7 x=\frac{7}{2} \Rightarrow x=\frac{7}{7 \times 2} \\ \end{aligned}

\Rightarrow x=\frac{1}{2}

Now x=\frac{1}{2} & y=\frac{1}{2} satisfies the third equation

Hence given vectors are coplanar

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