Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • The area of the quadrilateral ABCD with vertices A(2,1,1), B (1,2, 5), C(–2,–3, 5) and D (1, -6, -7) is equal toOption: 1 54</di

The area of the quadrilateral ABCD with vertices A(2,1,1), B (1,2, 5), C(–2,–3, 5) and D (1, -6, -7) is equal to

Option: 1

54


Option: 2

9 \sqrt{38}


Option: 3

48


Option: 4

8 \sqrt{38}


Answers (1)

best_answer

\begin{aligned} & \text { Vector Area }=\overrightarrow{\mathrm{v}} \\ & =\frac{1}{2} \overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}+\frac{1}{2} \overrightarrow{\mathrm{AC}} \times \overrightarrow{\mathrm{AD}} \\ & =\frac{1}{2}(\overrightarrow{\mathrm{AB}}-\overrightarrow{\mathrm{AD}}) \times \overrightarrow{\mathrm{AC}} \quad\left(\begin{array}{l} \overrightarrow{\mathrm{AB}}=-\hat{\mathrm{i}}+\hat{\mathrm{j}}+4 \hat{\mathrm{k}} \\ \overrightarrow{\mathrm{AD}}=-\hat{\mathrm{i}}-7 \hat{\mathrm{j}}-8 \hat{\mathrm{k}} \\ \overrightarrow{\mathrm{AC}}=-4 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \end{array}\right) \\ & =\frac{1}{2}(8 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}) \times(-4)(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}) \end{aligned}

\begin{aligned} & =\frac{1}{2}(8 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}) \times(-4)(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}) \\ & =\frac{1}{2}\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 0 & 8 & 12 \\ 1 & 1 & -1 \end{array}\right| \\ & =(-2)(-20 \hat{\mathrm{i}}+12 \hat{\mathrm{j}}-8 \hat{\mathrm{k}}) \\ & =8(5 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \\ & \therefore \text { Area }=|\overrightarrow{\mathrm{v}}|=8 \sqrt{25+9+4}=8 \sqrt{38} \text { Ans. } \end{aligned}

Posted by

Gunjita

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

Amrita BTech Admission 2025: JEE Main-based registration deadline extended to August 30

Latest Question