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  • Explain solution RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 8 maths

Explain solution RD Sharma class 12 chapter 26 Direction Cosines and Direction Ratios exercise Fill in the blanks question 8 maths

Answers (1)

Final Answer :  \gamma=\frac{\pi}{3}

Hint:

Use property of direction cosine  l^{2}+m^{2}+n^{2}=1

Given:

\alpha=\frac{\pi}{4}, \beta=\frac{\pi}{3}

To Find: 

 \gamma = angle with z-axis

Solution:

Since line makes angle \frac{\pi}{4} \text { and } \frac{\pi}{3}  with x and y axes, there direction cosines are

l=\cos \alpha, m=\cos \beta \text { and } n=\cos \gamma

\begin{aligned} &\text { As } l^{2}+m^{2}+n^{2}=1 \\\\ &\therefore \cos ^{2} \frac{\pi}{4}+\cos ^{2} \frac{\pi}{3}+\cos ^{2} \gamma=1 \end{aligned}                \left[\cos \left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}, \cos \left(\frac{\pi}{3}\right)=\frac{1}{2}\right]

\begin{aligned} &\frac{1}{2}+\frac{1}{4}+\cos ^{2} \gamma=1 \\\\ &\cos ^{2} \gamma=1-\frac{3}{4} \\\\ &\cos ^{2} \gamma=\frac{1}{4} \end{aligned}

\begin{aligned} &\cos \gamma=\frac{1}{2} \\\\ &\cos \gamma=\cos \frac{\pi}{3} \quad\left[\cos \frac{\pi}{3}=\frac{1}{2}\right] \end{aligned}

\gamma=\frac{\pi}{3}

Therefore, the angle made with z -axls is \frac{\pi }{3}

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