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  • Need Solution for R.D.Sharma Maths Class 12 Chapter 23 Scalar or Dot Products Exercise 23.1 Question 34 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 23 Scalar or Dot  Products  Exercise 23.1 Question 34 Maths Textbook Solution.

Answers (1)

Answer:  6\hat{i}+2\hat{k},-\hat{i}-2\hat{j}+3\hat{k}

Hint: You must know the rules of solving vectors.

Given:   Express the vector \vec{a}=5\hat{i}-2\hat{j}+5\hat{k}  as the sum of the two vectors such that one is parallel to vector \vec{b}=3\hat{i}+\hat{k} and other is perpendicular to \vec{b}.

Solution: Given that,

\vec{a}=5\hat{i}-2\hat{j}+5\hat{k} and \vec{b}=3\hat{i}+\hat{k}

Let \vec{x}. and \vec{y}. be such that

\begin{aligned} &\vec{a}=\vec{x}+\vec{y} \\\\ &\vec{y}=\vec{a}-\vec{x} \rightarrow(1) \\ \end{aligned}

Since, \vec{x}.is parallel to \vec{b}.

\begin{aligned} &\vec{x}=t(3 \hat{\imath}+\hat{k}) \\\\ &\vec{x}=3 t \hat{\imath}+t \hat{k} \\ \end{aligned}

Substituting the values, \vec{x}. and \vec{a}.

\begin{aligned} &\vec{y}=5 \hat{\imath}-2 \hat{\jmath}+5 \hat{k}-3 t \hat{\imath}+t \hat{k} \\\\ &\vec{y}=(5-3 t) \hat{\imath}-2 \hat{\jmath}+(5-t) \hat{k} \\ \end{aligned}

Since \vec{y}. is perpendicular to  \vec{b}.

\begin{aligned} &\vec{y} \cdot \vec{b}=0 \\ &{[(5-3 t) \hat{\imath}-2 \hat{\jmath}+(5-t) \hat{k}] \cdot 3 \hat{\imath}+\hat{k}=0} \\ &3(5-3 t)+0+(5-t)=0 \\ &15-9 t+5 t=0 \\ &20-10 t=0 \\ &t=2 \end{aligned}

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