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  • Explain solution for RD Sharma maths class 12 chapter 26 Direction Cosines and Direction Ratios exercise 26.1 question 16 sub question 1 maths textbook solution

Explain solution for RD Sharma maths class 12 chapter 26 Direction Cosines and Direction Ratios exercise 26.1 question 16 sub question 1 maths textbook solution

Answers (1)

Answer: Angle between lines is \frac{\pi }{3}

Given: 

              \begin{aligned} &1+m+n=0 \\ &1^{2}+m^{2}+n^{2}=0 \end{aligned} . Find angle between the lines

Hint: Use \cos \theta=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}

Solution: we have

              I+m+n=0                                                 ………………. (1)

              m=-I-n                                                     ………………. (2)

            \begin{aligned} &\left.\Rightarrow\right|^{2}+m^{2}+n^{2}=0 \quad[m=-1-n] \\ &\left.\Rightarrow\right|^{2}+(-1-n)^{2}-n^{2}=0 \\ &\left.\Rightarrow\right|^{2}+\left.\right|^{2}+n^{2}+2 \mid n-n^{2}=0 \\ &\left.\Rightarrow 2\right|^{2}+2 \ln =0 \\ &\Rightarrow 2 l(1+n)=0 \end{aligned}

Either \mid=0 or \mid=-n

          If \mid=0, then equ (2), m=-n

          If \mid=-n$, then equ $(2), m=0

Thus, direction ratio of two lines are proportional to

         \begin{aligned} &(0,-n, n) \text { and }(-n, 0, n) \\ &\Rightarrow(0,-1,1) \text { and }(-1,0,1) \\ &\Rightarrow\left(a_{1}, b_{1}, c_{1}\right)=(0,-1,1) \\ &\Rightarrow\left(a_{2}, b_{2}, c_{2}\right)=(-1,0,1) \end{aligned}

Angle between two lines is

           

              \begin{aligned} &\cos \theta=\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}} \sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} \\ &\cos \theta=\frac{(0)(-1)+(-1)(0)+(1)(1)}{\sqrt{0^{2}+(-1)^{2}+(1)^{2}} \sqrt{(-1)^{2}+0^{2}+1^{2}}} \\ &\cos \theta=\frac{1}{\sqrt{2} \sqrt{2}} \\ &\cos \theta=\frac{1}{2} \\ &\theta=\cos ^{-1}\left(\frac{1}{2}\right) \\ \end{aligned}

              \theta=\frac{\pi}{3}  

Therefore, angle between two lines is \frac{\pi}{3}       

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