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  • Please solve RD Sharma class 12 chapter Vector or Cross Product exercise 24.1 question 29 maths textbook solution

Please solve RD Sharma class 12 chapter Vector or Cross Product exercise 24.1 question 29 maths textbook solution

Answers (1)

Answer:

\frac{\sqrt{61}}{2} \text { square units }

Hint:

 To solve this we use area of triangle ABC

Given:

\begin{aligned} &A(2,3,5) \\ &B(3,5,8) \\ &C(2,7,8) \end{aligned}

\text { Area of } \Delta \mathrm{ABC}=\frac{1}{2}|\overrightarrow{A B} \times \overrightarrow{B C}|

Solution:

\begin{aligned} &\overrightarrow{A B}=\hat{i}+2 \hat{j}+3 \hat{k} \\\\ &\overrightarrow{B C}=-\hat{i}+2 \hat{j} \\\\ &\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} \\\\ &|\overrightarrow{A B} \times \overrightarrow{B C}|=\sqrt{(-6)^{2}+(-3)^{2}+(4)^{2}} \end{aligned}

\begin{aligned} &|\overrightarrow{A B} \times \overrightarrow{B C}|=\sqrt{36+9+16} \\\\ &|\overrightarrow{A B} \times \overrightarrow{B C}|=\sqrt{61}\\\\ \end{aligned}

Area \; of \; \Delta \mathrm{ABC}=\frac{1}{2}|\sqrt{61}|

=\frac{\sqrt{61}}{2} square\; units

 

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