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  • Provide solution for RD Sharma Maths Class 12 Chapter 24 Vector or Cross Product exercise Very Short Answer Question, question 13.

Provide solution for RD Sharma Maths Class 12 Chapter 24 Vector or Cross Product exercise Very Short Answer Question, question 13.

Answers (1)

ANSWER: \overrightarrow{0}

 

HINT:

Cross product the value and then do dot product

GIVEN:

 Two vectors \overrightarrow{a}\; \text{and}\: \overrightarrow{b} is given

SOLUTION:

  Let

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

Now, 

\begin{aligned} &\operatorname{For}(\vec{a} \times \vec{b}) \cdot \vec{b} \\ &\begin{aligned} (\vec{a} \times \vec{b}) &=\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} \end{aligned}

Now for 

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

Hence, \overrightarrow{0}

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