If A(1,-1,2), B(5,7,-6), C(3,4,-10) and D(-1,-4,-2) are the vertices of a quadrilateral A B C D , then its area is :

Correct answer: B. 12 29 aligned & A(1,-1,2) & B(5,7,-6) & C(3,4,-10) & D(-1,-4,-2) & Area =(1)/(2)| AC × BD |=(1)/(2)|(2 i +5 j -12 k ) ×(6 i +11 j -4 k )| & =(1)/(2)|112 i -64 j -8 k | & =4|14 i -8 j - k | & =4 196+64+1 & =4 261 & =12 29 aligned

JEE Main 20245 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 5 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Straight Lines

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Straight Lines

If A(1,-1,2), B(5,7,-6), C(3,4,-10) and D(-1,-4,-2) are the...

\mathrm{A}(1,-1,2), \mathrm{B}(5,7,-6), \mathrm{C}(3,4,-10)

and

\mathrm{D}(-1,-4,-2)

are the vertices of a quadrilateral

A B C D

, then its area is :

Options

A $48 \sqrt{7}$
B $12 \sqrt{29}$
C $24 \sqrt{7}$
D $24 \sqrt{29}$

Correct answer

B. $12 \sqrt{29}$

Step-by-step solution

\begin{aligned} & \mathrm{A}(1,-1,2) \\ & \mathrm{B}(5,7,-6) \\ & \mathrm{C}(3,4,-10) \\ & \mathrm{D}(-1,-4,-2) \\ & \text{ Area }=\frac{1}{2}|\overrightarrow{\mathrm{AC}} \times \overrightarrow{\mathrm{BD}}|=\frac{1}{2}|(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-12 \hat{\mathrm{k}}) \times(6 \hat{\mathrm{i}}+11 \hat{\mathrm{j}}-4 \hat{\mathrm{k}})| \\ & =\frac{1}{2}|112 \hat{\mathrm{i}}-64 \hat{\mathrm{j}}-8 \hat{\mathrm{k}}| \\ & =4|14 \hat{\mathrm{i}}-8 \hat{\mathrm{j}}-\hat{\mathrm{k}}| \\ & =4 \sqrt{196+64+1} \\ & =4 \sqrt{261} \\ & =12 \sqrt{29}\end{aligned}

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