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  • Need solution for RD Sharma Maths Class 12 Chapter 24 Vector Or Cross Product Exercise Very Short Answer Question, question 22.

Need solution for RD Sharma Maths Class 12 Chapter 24 Vector Or Cross Product Exercise Very Short Answer Question, question 22.

Answers (1)

ANSWER:

  0

HINT:

Simplify  \overrightarrow{a}\times \widehat{i}=\widehat{j} and then find \overrightarrow{a}. \widehat{i}

GIVEN:

\overrightarrow{a} is a unit vector

SOLUTION:

\begin{aligned} \text { so }|\vec{a}| &=1 \\ \text { Let, } \vec{a} &=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k} \\ \vec{a} \times \hat{i} &=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ 1 & 0 & 0 \end{array}\right| \\ &=\hat{i}(0-0)-\hat{j}\left(0-a_{3}\right)+\hat{k}\left(0-a_{2}\right) \\ \vec{a} \times \hat{i} &=a_{3} \hat{j}-a_{2} \hat{k} \\ \text { Given, } \vec{a} \times \hat{i} &=\hat{j} \\ & \therefore \hat{j}=a_{3} \hat{j}-a_{2} \hat{k} \end{aligned}

i.e.,

0 \hat{i}+\hat{j}+0 \hat{k}=a_{3} \hat{j}-a_{2} \hat{k}+0 \hat{i}

Comparing co-efficient of \widehat{i},\widehat{j}\text{and } \widehat{k}

\begin{aligned} &a_{3}=1 \\ &a_{2}=0 \\ &a_{1}=0 \\ &\therefore \vec{a}=a \hat{i}+0+\hat{k} \\ &\vec{a}=0+\hat{k} \\ &\therefore \vec{a} .=\hat{k} \end{aligned}

\begin{aligned} \text { Now, } \vec{a} \hat{i} &=\hat{k} \hat{i} \\ &=0 \quad[\because \text { Dot product is zero, } \hat{i} \hat{j}=\hat{k} \hat{i}] \end{aligned}

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