Question

If \(\vec p = \frac{{\left( {\vec b \times \vec c} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}},\vec q = \frac{{\left( {\vec c \times \vec a} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}},\vec r = \frac{{\left( {\vec a \times \vec b} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}}\) where a, b, c are three non-coplanar vectors, then the value of \(\left( {a + b + c} \right).\left( {p + q + r} \right)\)is given by [MNR\(1992\); UPSEAT\(2000\)].
A) \[3\]
 B) \[2\]
C) \[1\]
D) \[0\]

Answer 1

Hint: In this question we are going to use the algebra of vectors. Use vector multiplication and addition. Solve each part of the equation individually and after that add all these to get required value. Non coplanar vectors are those vectors which are not present in the same plane or parallel planes.If two vectors in a scalar triple product are the same then the scalar triple product will be zero.



Formula Used:Algebra of vectors i.e. vector addition and vector multiplication is used in this question.
\(\left( {\vec a + \vec b + \vec c} \right).\left( {\vec p + \vec q + \vec r} \right) = \vec a.\vec p + \vec a.\vec q + \vec a.\vec r + \vec b.\vec p + \vec b.\vec q + \vec b.\vec r + \vec c.\vec p + \vec c.\vec q + \vec c.\vec r\)



Complete step by step solution:Given: Three vectors p, q and r are non coplanar and \(\vec p = \frac{{\left( {\vec b \times \vec c} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}},\vec q = \frac{{\left( {\vec c \times \vec a} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}},\vec r = \frac{{\left( {\vec a \times \vec b} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}}\)
\(\left( {\vec a + \vec b + \vec c} \right).\left( {\vec p + \vec q + \vec r} \right) = \vec a.\vec p + \vec a.\vec q + \vec a.\vec r + \vec b.\vec p + \vec b.\vec q + \vec b.\vec r + \vec c.\vec p + \vec c.\vec q + \vec c.\vec r\)
\(\vec a.\vec q = a.\left( {\frac{{\vec c \times \vec a}}{{\vec a\vec b\vec c}}} \right) = \frac{{\left[ {\vec a\vec c\vec a} \right]}}{{\left[ {\vec a\vec b\vec c} \right]}} = 0\)

If two vectors in a scalar triple product are the same then the scalar triple product will be zero.
Similarly some other scalar dot product will also be zero:
\(\vec a.\vec r = \vec b.\vec p = \vec b.\vec r = \vec c.\vec p = \vec c.\vec q = 0\)
\(\left( {\vec a + \vec b + \vec c} \right).\left( {\vec p + \vec q + \vec r} \right) = \vec a.\vec p + \vec b.\vec q + \vec c.\vec r\)
\(\vec a.\vec p + \vec b.\vec q + \vec c.\vec r = \vec a.\frac{{\left( {\vec b \times \vec c} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}} + \vec b.\frac{{\left( {c \times \vec a} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}} + \vec c.\frac{{\left( {\vec a \times \vec b} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}}\)
\(\vec a.\frac{{\left( {\vec b \times \vec c} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}} + \vec b.\frac{{\left( {c \times \vec a} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}} + \vec c.\frac{{\left( {\vec a \times \vec b} \right)}}{{\left[ {\vec a\vec b\vec c} \right]}} = \frac{{\left[ {\vec a\vec b\vec c} \right]}}{{\left[ {\vec a\vec b\vec c} \right]}} + \frac{{\left[ {\vec a\vec b\vec c} \right]}}{{\left[ {\vec a\vec b\vec c} \right]}} + \frac{{\left[ {\vec a\vec b\vec c} \right]}}{{\left[ {\vec a\vec b\vec c} \right]}} = 3\)



Option ‘A’ is correct

Note:Here in this question we have to find the value of the given vector equation. We will use the algebra of vectors i.e. multiplication and addition of vectors in order to find required value.
Vector addition follows commutative and associative law.
Triangle law and parallelogram law are two methods of vector addition.
Vector products are distributive in nature.
Don’t try to solve the whole equation together because it will take more time and there will be a chance of mistakes.