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  • Provide solution for RD Sharma maths class 12 chapter Vector or Cross Product exercise 24.1 question 34

Provide solution for RD Sharma maths class 12 chapter Vector or Cross Product exercise 24.1 question 34

Answers (1)

Answer:

 \sqrt{61} Square units

Hint:

To solve this we use area of triangle

Given:

A(1,1,2) ; B(2,3,5) ; C(1,5,5)

Solution:

\begin{aligned} &D=\frac{1}{2}(\overrightarrow{A B} \times \overrightarrow{B C}) \\\\ &\Rightarrow \overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A} \\\\ &=(2 \hat{i}+3 \hat{j}+5 \hat{k})-(\hat{i}+\hat{j}+2 \hat{k}) \\\\ &=\hat{i}+2 \hat{j}+3 \hat{k} \end{aligned}

\begin{aligned} &\Rightarrow \overrightarrow{B C}=\overrightarrow{O C}-\overrightarrow{O B} \\\\ &=(\hat{i}+5 \hat{j}+5 \hat{k})-(2 \hat{i}+3 \hat{j}+5 \hat{k}) \\\\ &=-\hat{i}+2 \hat{j} \end{aligned}

\begin{aligned} &D=\frac{1}{2}(\overrightarrow{A B} \times \overrightarrow{B C}) \\\\ &=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ -1 & 2 & 0 \end{array}\right| \end{aligned}

\begin{aligned} &=\hat{i}(0-6)+\hat{j}(0-(-3))+\hat{k}(2-(-2)) \\\\ &=-6 \hat{i}-3 \hat{j}+4 \hat{k} \end{aligned}

\begin{aligned} &=(\overrightarrow{A B} \times \overrightarrow{B C})=\sqrt{(-6)^{2}+(-3)^{2}+(4)^{2}} \\\\ &=\sqrt{36+9+6} \\\\ &=\sqrt{61} \text { square units } \end{aligned}

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