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  • Provide Solution For R. D. Sharma Maths Class 12 Chapter 23 Scalar or Dot Products Exercise 23.1 Question 16 Maths Textbook Solution.

Provide Solution For R. D. Sharma Maths Class 12 Chapter 23 Scalar or Dot Products  Exercise 23.1 Question 16 Maths Textbook Solution.

Answers (1)

Answer: \lambda =\pm 1

Hint: you must know the rules of solving vectors

Given: if \vec{p}=5\hat{i}+\lambda \hat{j}-3\hat{k}   and  \vec{q}=\hat{i}+3\hat{j}-5\hat{k}

Then find the value of \lambda, so \vec{p}+\vec{q}and \vec{p}-\vec{q}are perpendicular vectors

Solution: given that

\begin{aligned} &\vec{p}=5 \hat{\imath}+\lambda \hat{\jmath}-3\hat{k}\text { and } \\ &\vec{q}=\hat{\imath}+3 \hat{\jmath}-5 \hat{k} \\ &\vec{p}+\vec{q}=(5 \hat{\imath}+\lambda \hat{\jmath}-3 \hat{k})+(\hat{\imath}+3 \hat{j}-5 \hat{k}) \end{aligned}

\begin{aligned} &=6 \hat{\imath}+(\lambda+3) \hat{\jmath}-8 \hat{k} \\ \end{aligned}

\begin{aligned} &\vec{p}-\vec{q}=(5 \hat{\imath}+\lambda \hat{\jmath}-3 \hat{k})-(\hat{\imath}+3 \hat{\jmath}-5 \hat{k}) \\ &=4 \hat{\imath}+(\lambda-3) \hat{\jmath}+2 \hat{k} \end{aligned}

Given that is \vec{p}+\vec{q}orthogonal to\vec{p}-\vec{q}

\begin{aligned} &\Rightarrow(\vec{p}+\vec{q})(\vec{p}-\vec{q})=0 \\\\&\Rightarrow[6 \hat{\imath}+(\lambda+3) \hat{\jmath}-8 \hat{k}] \cdot[4 \hat{\imath}+(\lambda-3) \hat{\jmath}+2 \hat{k}]=0 \\ \end{aligned}

\begin{aligned} &\Rightarrow 24+\lambda^{2}-9-16=0 \\\\ &\lambda^{2}=1 \\\\ &\therefore \lambda=\pm 1 \end{aligned}

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