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  • Explain solution for RD Sharma maths class 12 chapter 26 Direction Cosines and Direction Ratios exercise 26.1 question 11 maths textbook solution

Explain solution for RD Sharma maths class 12 chapter 26 Direction Cosines and Direction Ratios exercise 26.1 question 11 maths textbook solution

Answers (1)

Answer: Both the lines are perpendicular to each other

Hint: Angle between perpendicular line is 90^{\circ}

Given: \mathrm{A}(1,-1,2), \mathrm{B}(3,4,-2), \mathrm{C}(0,3,2) \& \mathrm{D}(3,5,6). Show AB is perpendicular to CD

Solution: we have \mathrm{A}(2,3,6), \mathrm{B}(1,2,2), \mathrm{C}(0,3,2) \& \mathrm{D}(3,5,6)

Let \theta be the angle between two lines whose direction cosines are \left ( a_{1},b_{1},c_{1} \right ) &  \left ( a_{2},b_{2},c_{2} \right )

Direction ratio of AB=\left ( 3-1,4-\left ( -1 \right ),-2-2 \right )

              \left ( a_{1},b_{1},c_{1} \right )=\left ( 2,5,-4 \right )

Direction ratio of CD=\left ( 3-0,5-3,6-2 \right )

              \left ( a_{2},b_{2},c_{2} \right )=\left ( 3,2,4 \right )

Now,

              \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{(2)(3)+(5)(2)+(-4)(4)}{\sqrt{2^{2}+(5)^{2}+(-4)^{2}} \sqrt{3^{2}+(2)^{2}+4^{2}}} \\ &\cos \theta=\frac{6+10-16}{\sqrt{45} \sqrt{29}} \\ &\cos \theta=0 \\ &\theta=\cos ^{-1}(0) \\ &\theta=90^{\circ} \end{aligned}

Therefore, AB and CD are perpendicular to each other.

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