Question

A tetrahedron has vertices $P(1,2,1),Q(2,1,3),R( - 1,1,2)$ and $O(0,0,0)$. The angle between the faces OPQ and PQR is:
A ${\cos ^{ - 1}}\left( {\dfrac{9}{{35}}} \right)$
B ${\cos ^{ - 1}}\left( {\dfrac{{19}}{{35}}} \right)$
C ${\cos ^{ - 1}}\left( {\dfrac{{17}}{{31}}} \right)$
D ${\cos ^{ - 1}}\left( {\dfrac{7}{{31}}} \right)$

Answer 1

Hint: In this question we have been given a tetrahedron PQRO with vertices, $P(1,2,1),Q(2,1,3),R( - 1,1,2)$and $O(0,0,0)$ we need to find out the angle between the faces OPQ and PQR. For that we need to find the cross product, and after that use the formula:$\cos \theta = \left( {\dfrac{{a.b}}{{\left| a \right|.\left| b \right|}}} \right)$, where $a = \mathop {OP}\limits^ \to \times \mathop {OQ}\limits^ \to $and $\mathop {b = PQ}\limits^ \to \times \mathop {PR}\limits^ \to $

Complete step-by-step answer:
We have been provided with a tetrahedron PQRO with vertices $P(1,2,1),Q(2,1,3),R( - 1,1,2)$and $O(0,0,0)$,
So, according to the question we need to find the angle between the faces OPQ and PQR. So, for that firstly we need to find the vectors OP and OQ.
$\begin{gathered}
  \mathop {OP}\limits^ \to = (\hat i(1 - 0) + \hat j(2 - 0) + \hat k(1 - 0)) \\
  \mathop {OQ}\limits^ \to = (\hat i(2 - 0) + \hat j(1 - 0) + \hat k(3 - 0)) \\
\end{gathered} $,
Simplifying it further we get,
$\begin{gathered}
  \mathop {OP}\limits^ \to = (1\hat i + 2\hat j + 1\hat k) \\
  \mathop {OQ}\limits^ \to = (2\hat i + 1\hat j + 3\hat k) \\
\end{gathered} $
Now we will be calculating the cross product of:
 $\begin{gathered}
  \mathop {OP}\limits^ \to = (1\hat i + 2\hat j + 1\hat k) \\
  \mathop {OQ}\limits^ \to = (2\hat i + 1\hat j + 3\hat k) \\
\end{gathered} $,
So, the cross product: $\mathop {OP}\limits^ \to \times \mathop {OQ}\limits^ \to = \left| {\left( {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  1&2&1 \\
  2&1&3
\end{array}} \right)} \right|$,
Now we will be finding the cross product by solving the above determinant: $\mathop {OP}\limits^ \to \times \mathop {OQ}\limits^ \to = \hat i(6 - 1) - \hat j(3 - 2) + \hat k(1 - 4)$,
The cross product comes out to be: $\mathop {OP}\limits^ \to \times \mathop {OQ}\limits^ \to = 5\hat i - \hat j - 3\hat k$,
Now we will be finding the vectors PQ and PR.
$\begin{gathered}
  \mathop {PQ}\limits^ \to = (\hat i(2 - 1) + \hat j(1 - 2) + \hat k(3 - 1)) \\
  \mathop {PR}\limits^ \to = (\hat i( - 1 - 2) + \hat j(1 - 2) + \hat k(2 - 1)) \\
\end{gathered} $
Now we will be simplifying them further,
$\begin{gathered}
  \mathop {PQ}\limits^ \to = (\hat i - \hat j + 2\hat k) \\
  \mathop {PR}\limits^ \to = ( - 2\hat i - \hat j + \hat k) \\
\end{gathered} $
Now we will be calculating the cross product of:
$\begin{gathered}
  \mathop {PQ}\limits^ \to = (\hat i - \hat j + 2\hat k) \\
  \mathop {PR}\limits^ \to = ( - 2\hat i - \hat j + \hat k) \\
\end{gathered} $,

So, the cross product: $b = \mathop {PQ}\limits^ \to \times \mathop {PR}\limits^ \to $
$\mathop {PQ}\limits^ \to \times \mathop {PR}\limits^ \to = \left| {\left( {\begin{array}{*{20}{c}}
  {\hat i}&{\hat j}&{\hat k} \\
  1&{ - 1}&2 \\
  { - 2}&{ - 1}&1
\end{array}} \right)} \right|$,
Now we will be finding the cross product by solving the above determinant:$\mathop {PQ}\limits^ \to \times \mathop {PR}\limits^ \to = \hat i( - 1 + 2) - \hat j(1 + 4) + \hat k( - 1 - 2)$,
The cross product comes out to be: $\mathop {PQ}\limits^ \to \times \mathop {QR}\limits^ \to = \hat i - 5\hat j - 3\hat k$,
Now as we want to calculate the angle between OPQ and PQR,
So, we will be using the formula:$\cos \theta = \left( {\dfrac{{a.b}}{{\left| a \right|.\left| b \right|}}} \right)$,
In this question, $a = \mathop {OP}\limits^ \to \times \mathop {OQ}\limits^ \to $ and $b = \mathop {PQ}\limits^ \to \times \mathop {QR}\limits^ \to $,
So, we will be putting the values of a and b in $\cos \theta = \left( {\dfrac{{a.b}}{{\left| a \right|.\left| b \right|}}} \right)$,
So, the equation becomes: $\cos \theta = \left( {\dfrac{{\left( {\mathop {OP}\limits^ \to \times \mathop {OQ}\limits^ \to } \right).\left( {\mathop {PQ}\limits^ \to \times \mathop {QR}\limits^ \to } \right)}}{{\left| {\mathop {OP}\limits^ \to \times \mathop {OQ}\limits^ \to } \right|.\left| {\mathop {PQ}\limits^ \to \times \mathop {QR}\limits^ \to } \right|}}} \right)$,
Putting the values, we will get: $\cos \theta = \left( {\dfrac{{(5\hat i - \hat j - 3\hat k).\left( {\hat i - 5\hat j - 3\hat k} \right)}}{{\left| {5\hat i - \hat j - 3\hat k} \right|.\left| {\hat i - 5\hat j - 3\hat k} \right|}}} \right)$,
Now we need to calculate the dot product of: $(5\hat i - \hat j - 3\hat k).\left( {\hat i - 5\hat j - 3\hat k} \right)$,
So, the dot product: $5\hat i - \hat j - 3\hat k).\left( {\hat i - 5\hat j - 3\hat k} \right) = 5(1) - 1( - 5) - 3( - 3)$,
The dot product comes out to be: $5\hat i - \hat j - 3\hat k).\left( {\hat i - 5\hat j - 3\hat k} \right) = 19$,
Now we need to calculate the value of $\left| {5\hat i - \hat j - 3\hat k} \right|$,
So, it will be: $\left| {5\hat i - \hat j - 3\hat k} \right|$=$\sqrt {({5^2}) + ( - {1^2}) + ( - {3^2})} = \sqrt {35} $,
Now we need to calculate the value of $\left| {\hat i - 5\hat j - 3\hat k} \right|$,
So, it will be: $\left| {\hat i - 5\hat j - 3\hat k} \right|$=$\sqrt {({1^2}) + ( - {5^2}) + ( - {3^2})} = \sqrt {35} $
Keeping this value in $\cos \theta = \left( {\dfrac{{(5\hat i - \hat j - 3\hat k).\left( {\hat i - 5\hat j - 3\hat k} \right)}}{{\left| {5\hat i - \hat j - 3\hat k} \right|.\left| {\hat i - 5\hat j - 3\hat k} \right|}}} \right)$,
The angle between OPQ and PQR: $\cos \theta = \left( {\dfrac{{19}}{{\sqrt {35} .\sqrt {35} }}} \right)$
So, the angle comes out to be: $\begin{gathered}
  \cos \theta = \left( {\dfrac{{19}}{{35}}} \right) \\
  \theta = {\cos ^{ - 1}}\left( {\dfrac{{19}}{{35}}} \right) \\
\end{gathered} $

So, the correct answer is “Option b”.

Note: In this question do remember that during dot product does not include the direction vectors, because it is a scalar product. While finding the cross product of OP and OQ, PQ and PR find the vectors $\mathop {OP}\limits^ \to ,\mathop {OQ}\limits^ \to ,\mathop {PQ}\limits^ \to ,\mathop {PR}\limits^ \to $ first do avoid any kind of mistakes.