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  • Provide solution for RD Sharma Maths Class 12 Chapter 26 Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 11.

Provide solution for RD Sharma Maths Class 12 Chapter 26 Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 11.

Answers (1)

Final answer: (a) (-1,2,-2)

Hint: Use direction to find direction cosine

Given:

|\overrightarrow{O P}|=3 \&(a, b, c)=(-1,2,-2)

To Find: Point P

Solution: Direction cosine is related to direction ratio as

\begin{aligned} &l=\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, m=\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, n=\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}} \\ &l=\frac{-1}{\sqrt{(-1)^{2}+2^{2}+(-2)^{2}}}, m=\frac{2}{\sqrt{(-1)^{2}+2^{2}+(-2)^{2}}}, n=\frac{-2}{\sqrt{(-1)^{2}+2^{2}+(-2)^{2}}} \\ &l=-\frac{1}{3}, m=\frac{2}{3}, n=-\frac{2}{3} \end{aligned}

Now,

\overrightarrow{OP}

=Position of P- Position of O

\begin{aligned} &\overrightarrow{O P}=(x \hat{i}+y \hat{j}+z \hat{k})-(0 \hat{i}+0 \hat{j}+0 \hat{k}) \\ &\overrightarrow{O P}=(x \hat{i}+y \hat{j}+z \hat{k}) \end{aligned}

where (x,y,z) are the coordinates of P

Now,

\begin{aligned} &\overrightarrow{O P}=|\overrightarrow{O P}|\left(l^{\Lambda}+m \hat{j}+n \hat{k}\right) \\ &(x \hat{i}+y \hat{j}+z \stackrel{\Lambda}{k})=3\left(\frac{-1}{3} \hat{i}+\frac{2}{3} j-\frac{2}{3} \hat{k}\right) \\ &\left(x \stackrel{\Lambda}{i}+y \hat{j}+z^{\Lambda}\right)=(-\hat{i}+2 \hat{j}-2 \hat{k}) \end{aligned}

Comparing both,

(x,y,z)=(-1,2,-2)

Therefore coordinates of point P are (-1,2,-2). Hence option (a) is correct

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