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  • Please solve RD Sharma Class 12 Chapter 24 Vector or Cross Product Exercise Very Short Answer Question, question 10 Maths textbook solution.

Please solve RD Sharma Class 12 Chapter 24 Vector or Cross Product Exercise Very Short Answer Question, question 10 Maths textbook solution.

Answers (1)

ANSWER:

  0

HINT:

  Find the cross product and then the dot product

GIVEN:

 Two vectors \overrightarrow{a} and \overrightarrow{b}                                              

SOLUTION:

Let

\begin{aligned} &\vec{a}=a_{1} \hat{i}+b \hat{j}+c_{1} \hat{k} \\ &\vec{b}=a_{2} \hat{i}+b_{2} \hat{j}+c_{2} \hat{k} \end{aligned}

Now,

For, \overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{a})

We calculate

\begin{aligned} (\vec{b} \times \vec{a}) &=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right| \\ &=\hat{i}\left(b_{1} c_{2}-b_{2} c_{1}\right)-\hat{j}\left(a_{1} c_{2}-a_{2} c_{1}\right)+\hat{k}\left(a_{1} b_{2}-a_{2} b_{1}\right) \end{aligned}

Now,

$$ \begin{aligned} &\text { For } \vec{a} \cdot(\vec{b} \times \vec{a}) \\ &=\left(a_{1} \hat{i}+b \hat{j}+c_{1} \hat{k}\right) \cdot\left[\hat{i}\left(b_{1} c_{2}-b_{2} c_{1}\right)-\hat{j}\left(a_{1} c_{2}-a_{2} c_{1}\right)+\hat{k}\left(a_{1} b_{2}-a_{2} b_{1}\right)\right] \\ &=a_{1}\left(b_{1} c_{2}-b_{2} c_{1}\right)-b_{1}\left(a_{1} c_{2}-a_{2} c_{1}\right)+c_{1}\left(a_{1} b_{2}-a_{2} b_{1}\right) \\ &=a_{1} b_{1} c_{2}-a_{1} b_{2} c_{1}-a_{1} b_{1} c_{2}+a_{2} b_{1} c_{1}+a_{1} b_{2} c_{1}-a_{2} b_{1} c_{1} \\ &=0 \end{aligned}

Thus, 

\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{a})=0 

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