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  • Please Solve RD Sharma Class 12 Chapter 27 Straight Line in Space Exercise Fill in the blanks Question 15 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 27 Straight Line in Space Exercise Fill in the blanks Question 15 Maths Textbook Solution.

Answers (1)

Answer : The value of k is  2

Hint :   The condition of coplanar is

             (\overrightarrow{a 2}-\overrightarrow{a 1}) \cdot(\overrightarrow{b 2} \times\overrightarrow{b 1})=0

Given :  The lines are ,

            \overrightarrow{\mathrm{r}}=2 \overrightarrow{\mathrm{a} 1}-3 \overrightarrow{\mathrm{a} 2}+\lambda \overrightarrow{\mathrm{b} 1}       -------1

            \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a} 2}+\mu \overrightarrow{\mathrm{b} 2}     ------2

Solution :  From the condition of co planarity

            \begin{aligned} &(\overrightarrow{a 2}-\overrightarrow{a 1}) \cdot(\overrightarrow{b 2} \times \overrightarrow{b 1})=0 \\ \end{aligned}

          \Rightarrow (\overrightarrow{a 2}-2 \vec{a} 1+3 \overrightarrow{a 2}) \cdot(\overrightarrow{b 2} \times \overrightarrow{b 1})=0 \\

          \Rightarrow (4 \vec{a} 2-2 \vec{a}) \cdot(\overrightarrow{b 2} \times \overrightarrow{b 1})=0

          \Rightarrow 2(2 \overrightarrow{\mathrm{a} 2}-\overrightarrow{\mathrm{a} 1}) \cdot \overrightarrow{\mathrm{b} 2} \times \overrightarrow{\mathrm{b} 1})=0 \\

          \Rightarrow(2 \overrightarrow{\mathrm{a} 2}-\overrightarrow{\mathrm{a} 1}) \cdot(\overrightarrow{\mathrm{b} 2} \times \overrightarrow{\mathrm{b} 1})=0 \\

          \Rightarrow 2 \overrightarrow{\mathrm{a} 2} \cdot(\overrightarrow{\mathrm{b} 2} x \overrightarrow{\mathrm{b} 1})-\overrightarrow{\mathrm{a} 1} \cdot(\overrightarrow{\mathrm{b} 2} x \overrightarrow{\mathrm{b} 1})=0 \\

         \Rightarrow 2[\overrightarrow{\mathrm{a} 2 \mathrm{~b} 1 \mathrm{~b} 2}]=[\overrightarrow{\mathrm{a} 1 \mathrm{~b} 1 \mathrm{~b} 2}]

         a \cdot(b \times c)=\left|\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right|=0                    [  If a,b,c are coplanar  ]

         \begin{aligned} &\Rightarrow \quad[\overrightarrow{a 1}\overrightarrow{b1}\overrightarrow{b2}]=2[\overrightarrow{a 2} \overrightarrow{b1}\overrightarrow{b 2}] \\ & \end{aligned}

         \Rightarrow \quad k=2

           So , the value of  k= 2

Posted by

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