Join the Most Relevant JEE Main 2025 Test Series & get 99+ percentile! Join Now
Search any question & find its solution
Question: Answered & Verified by Expert
A tetrahedron has vertices $O(0,0,0), A(1,2,1)$, $B(2,1,3), C(-1,1,2)$. If $\theta$ is the angle between the faces $O A B$ and $A B C$, then $\cos \theta=$
MathematicsThree Dimensional GeometryTS EAMCETTS EAMCET 2020 (10 Sep Shift 1)
Options:
  • A $\frac{1}{\sqrt{2}}$
  • B $\frac{19}{35}$
  • C $\frac{\sqrt{3}}{2}$
  • D $\frac{17}{31}$
Solution:
1271 Upvotes Verified Answer
The correct answer is: $\frac{19}{35}$
Equation of plane $O A B$ is given by,
$\begin{aligned}
& \left|\begin{array}{lll}
x-0 & y-0 & z-0 \\
1-0 & 2-0 & 1-0 \\
2-0 & 1-0 & 3-0
\end{array}\right|=0 \Rightarrow\left|\begin{array}{lll}
x & y & z \\
1 & 2 & 1 \\
2 & 1 & 3
\end{array}\right|=0 \\
& x(6-1)-y(3-2)+z(1-4)=0
\end{aligned}$


Equation of plane $A B C$ is given by,
$\begin{aligned}
\left|\begin{array}{ccc}
x-1 & y-2 & z-1 \\
2-1 & 1-2 & 3-1 \\
-1-1 & 1-2 & 2-1
\end{array}\right| & =0 \\
\left|\begin{array}{ccc}
x-1 & y-2 & z-1 \\
1 & -1 & 2 \\
-2 & -1 & 1
\end{array}\right| & =0
\end{aligned}$
$\begin{gathered}(x-1)(-1+2)-(y-2)(1+4)+(z-1)(-1-2)=0 \\ x-1-5 y+10-3 z+3=0\end{gathered}$

Then angle between planes represented by Eqs. (i) and (ii) is given by,
$\cos \theta=\frac{(5)(1)+(-1)(-5)+(-3)(-3)}{\sqrt{25+1+9} \sqrt{1+25+9}}=\frac{19}{35}$

Looking for more such questions to practice?

Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.