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  • Explain solution RD Sharma class 12 chapter Vector or Cross Product exercise 24.1 question 36 maths

Explain solution RD Sharma class 12 chapter Vector or Cross Product exercise 24.1 question 36 maths

Answers (1)

Answer:

\frac{2 \hat{i}-\hat{j}-\hat{k}}{\sqrt{6}},-\frac{3}{5} \hat{j}-\frac{4}{5} \hat{k}, \sqrt{404} \text { sq.units }

Hint:

To solve this we use determinant method

Given:

\begin{aligned} &\text { Magnitude }=10 \sqrt{3} \\ &\vec{a}=\hat{i}+2 \hat{j}+\hat{k} \\ &\vec{b}=-\hat{i}+3 \hat{j}+4 \hat{k} \end{aligned}

Solution:

\begin{aligned} &\vec{a}=\hat{i}+2 \hat{j}+\hat{k} \\\\ &\vec{b}=-\hat{i}+3 \hat{j}+4 \hat{k} \\\\ &\vec{a}+\vec{b}=\vec{c}=4 \hat{i}-2 \hat{j}-2 \hat{k} \end{aligned}

\begin{aligned} &|\vec{a} \times \vec{b}|=\frac{4 \hat{i}-2 \hat{j}-2 \hat{k}}{\sqrt{(4)^{2}+(-2)^{2}+(-2)^{2}}} \\\\ &=\frac{4 \hat{i}-2 \hat{j}-2 \hat{k}}{\sqrt{16+4+4}}=\frac{4 \hat{i}-2 \hat{j}-2 \hat{k}}{\sqrt{24}} \end{aligned}

\begin{aligned} &=\frac{4 \hat{i}-2 \hat{i}-2 \hat{k}}{2 \sqrt{6}} \\\\ &=\frac{2 \hat{i}-j-\hat{k}}{\sqrt{6}} \end{aligned}

Similarly

\begin{aligned} &\vec{d}=\vec{a}-\vec{b}=0 \hat{i}-6 \hat{j}-8 \hat{k} \\\\ &|\vec{a}-\vec{b}|=\frac{-6 \hat{j}-8 \hat{k}}{\sqrt{(-6)^{2}+(-8)^{2}}} \\\\ &=\frac{-6 \hat{j}-8 \hat{k}}{\sqrt{36+64}} \end{aligned}

\begin{aligned} &=\frac{-6 \hat{j}-8 \hat{k}}{10} \\\\ &=\frac{-3}{5} \hat{j}-\frac{4}{5} \hat{k} \end{aligned}

Area of the parallelogram =|\vec{a} \times \vec{b}|

=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & -4 & -5 \\ 2 & 2 & 3 \end{array}\right|

\begin{aligned} &=|-2 \hat{i}-16 \hat{j}+12 \hat{k}| \\\\ &=\sqrt{(-2)^{2}+(-16)^{2}+(12)^{2}} \\\\ &=\sqrt{4+256+144} \\\\ &=\sqrt{404} \text { sq.units } \end{aligned}

 

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infoexpert26

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