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  • Please solve RD Sharma class 12 chapter Vector or Cross Product exercise 24.1 question 33 maths textbook solution

Please solve RD Sharma class 12 chapter Vector or Cross Product exercise 24.1 question 33 maths textbook solution

Answers (1)

Answer:

Proved

Hint:

To solve this we use determinant method

Given:

\begin{aligned} &a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k} \\\\ &b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k} \\\\ &c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k} \end{aligned}

Solution:

\vec{b}+\vec{c}=\left(b_{1}+c_{1}\right) \hat{i}+\left(b_{2}+c_{2}\right) \hat{j}+\left(b_{3}+c_{3}\right) \hat{k}

\vec{a} \times(\vec{b}+\vec{c})=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ b_{1}+c_{1} & b_{2}+c_{2} & b_{3}+c_{3} \end{array}\right|

=\hat{i}\left[a_{2}\left(b_{3}+c_{3}\right)-a_{3}\left(b_{2}+c_{2}\right)\right]-\hat{j}\left[a_{1}\left(b_{3}+c_{3}\right)-a_{3}\left(b_{1}+c_{1}\right)\right]+\hat{k}\left[a_{1}\left(b_{2}+c_{2}\right)-a_{2}\left(b_{1}+c_{1}\right)\right]

\begin{aligned} &=\hat{i}\left(a_{2} b_{3}+a_{3} b_{2}\right)-\hat{j}\left(a_{1} b_{3}+a_{3} b_{1}\right)+\hat{k}\left(a_{1} b_{2}+a_{2} b_{1}\right)+\hat{i}\left(a_{2} c_{3}+a_{3} c_{2}\right)-\hat{j}\left(a_{1} c_{3}+a_{3} c_{1}\right)+\hat{k}\left(a_{1} c_{2}+a_{2} c_{1}\right) \\\\ &\vec{a} \times(\vec{b}+\vec{c})=(\vec{a} \times \vec{c})+(\vec{a} \times \vec{b}) \end{aligned}

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