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  • Provide Solution For R.D.Sharma Maths Class 12 Chapter 25 Scalar Triple Product Exercise 25.1 Question 5 Sub Question 4 Maths Textbook Solution.

Provide Solution For  R.D.Sharma Maths Class 12 Chapter 25 Scalar Triple Product Exercise 25.1 Question 5 Sub Question 4 Maths Textbook Solution.

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Answer :-  \lambda =\frac{-1}{3}

Hint :-  As it is given that vectors are coplanar. So use \left [ \vec{a}\: \: \vec{b}\: \: \vec{c} \right ]=0  to find \lambda

Given: \begin{aligned} &\vec{a}=\hat{\imath}+3 \hat{\jmath}=\hat{i}+3 \hat{\jmath}+0 \hat{k} \\ \end{aligned}

\begin{aligned} &\vec{b}=5 \hat{k}=0 \hat{\imath}+0 \hat{\jmath}+5 \hat{k} \\ &\vec{c}=\lambda \hat{\imath}-\hat{\jmath}=\lambda \hat{\imath}-\hat{\jmath}+0 \hat{k} \end{aligned}

If vectors are given coplanar then,

\left [ \vec{a}\: \: \vec{b}\: \: \vec{c} \right ]=0

\begin{aligned} &{\left[\begin{array}{ll} \vec{a}\ \vec{b} & \vec{c} \end{array}\right]=\left|\begin{array}{ccc} 1 & 3 & 0 \\ 0 & 0 & 5 \\ \lambda & -1 & 0 \end{array}\right|} \\ \end{aligned}

\begin{aligned} &=1(0+5)-3(0-5 \lambda)+0(0-0) \\ &=5+15 \lambda \\ \end{aligned}

As \left [ \vec{a}\: \: \vec{b}\: \: \vec{c} \right ]=0

\begin{aligned} &5+15 \lambda=0 \\ &\therefore \lambda=\frac{-5}{15} \\ &\lambda=\frac{-1}{3} \end{aligned}

    

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