Answer
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Hint: If a and b are two non-zero vectors and is the angle between them, their scalar product (or dot product) is denoted by \[a \cdot b\] and defined as the scalar \[\left| a \right|\left| b \right|{\rm{ }}cos\theta \], where |a| and |b| are moduli of a and b respectively and \[\theta \]. A scalar quantity is the dot product of two vectors.
Formula Used:The dot product of two vectors can be calculated as follows:
\[{\bf{a}}.\left( {{\bf{b}} + {\bf{c}}} \right) = {\bf{a}}.{\bf{b}} + {\bf{a}}.{\bf{c}}\]
Complete step by step solution:If a, b, and c are three vectors, their scalar triple product can be considered as the dot product of \[a\]and \[(b \times c)\]
It is commonly represented by
\[a \cdot \left( {b{\rm{ }} \times {\rm{ }}c} \right)\;\]
Or
\[\left[ {a{\rm{ }}b{\rm{ }}c} \right]\]
We have been given in the problem that,
\[a \cdot (b \times c)\]
According to the property of scalar triple product, we have
The value of the scalar triple product remains constant while ‘a’, ‘b’, and ‘c’ is cyclically permuted.
\[\left( {a{\rm{ }} \times {\rm{ }}b} \right) \cdot c = \left( {b{\rm{ }} \times {\rm{ }}c} \right) \cdot a\]
\[ = {\rm{ }}\left( {c{\rm{ }} \times {\rm{ }}a} \right).{\rm{ }}b\]
Or it can also be written as,
\[\left[ {a{\rm{ }}b{\rm{ }}c} \right]{\rm{ }} = {\rm{ }}\left[ {b{\rm{ }}c{\rm{ }}a} \right]{\rm{ }} = {\rm{ }}\left[ {c{\rm{ }}a{\rm{ }}b} \right]\]
The cyclic arrangement of vectors in a scalar triple product affects the sign but not the magnitude of the product.
That is, \[\left[ {a{\rm{ }}b{\rm{ }}c} \right] = - \left[ {b{\rm{ }}a{\rm{ }}c} \right]{\rm{ = }} - \left[ {c{\rm{ }}b{\rm{ }}a} \right] = - \left[ {a{\rm{ }}c{\rm{ }}b} \right]\]
The placements of the dot and cross in a scalar triple product can be swapped as long as the vectors' cyclic order remains constant.
That is, \[\left( {a{\rm{ }} \times {\rm{ }}b} \right) \cdot c{\rm{ }} = {\rm{ }}a \cdot \left( {b{\rm{ }} \times {\rm{ }}c} \right)\]
If any two of the vectors are equal, the scalar triple product of the three vectors is zero.
Therefore, the term \[a \cdot (b \times c)\] is equal to \[b \cdot \left( {c \times a} \right)\]
Option ‘C’ is correct
Note: The relation \[\left[ {a{\rm{ }}b{\rm{ }}c} \right] = 0\]is a necessary and sufficient condition for three non-zero non-collinear vectors to be coplanar.
The expression\[\left[ {{\rm{ }}a{\rm{ }}b{\rm{ }}c} \right] + \left[ {d{\rm{ }}c{\rm{ }}a} \right] + \left[ {d{\rm{ }}a{\rm{ }}b} \right] = \left[ {a{\rm{ }}b{\rm{ }}c} \right]\], four points with position vectors ‘a’, ‘b’, ‘c’, and ‘d’ are coplanar.
The volume of a parallelepiped with coterminous edges a, b, and c is \[\left[ {a{\rm{ }}b{\rm{ }}c} \right]\] or \[a\left( {b{\rm{ }} \times {\rm{ }}c} \right)\].
Formula Used:The dot product of two vectors can be calculated as follows:
\[{\bf{a}}.\left( {{\bf{b}} + {\bf{c}}} \right) = {\bf{a}}.{\bf{b}} + {\bf{a}}.{\bf{c}}\]
Complete step by step solution:If a, b, and c are three vectors, their scalar triple product can be considered as the dot product of \[a\]and \[(b \times c)\]
It is commonly represented by
\[a \cdot \left( {b{\rm{ }} \times {\rm{ }}c} \right)\;\]
Or
\[\left[ {a{\rm{ }}b{\rm{ }}c} \right]\]
We have been given in the problem that,
\[a \cdot (b \times c)\]
According to the property of scalar triple product, we have
The value of the scalar triple product remains constant while ‘a’, ‘b’, and ‘c’ is cyclically permuted.
\[\left( {a{\rm{ }} \times {\rm{ }}b} \right) \cdot c = \left( {b{\rm{ }} \times {\rm{ }}c} \right) \cdot a\]
\[ = {\rm{ }}\left( {c{\rm{ }} \times {\rm{ }}a} \right).{\rm{ }}b\]
Or it can also be written as,
\[\left[ {a{\rm{ }}b{\rm{ }}c} \right]{\rm{ }} = {\rm{ }}\left[ {b{\rm{ }}c{\rm{ }}a} \right]{\rm{ }} = {\rm{ }}\left[ {c{\rm{ }}a{\rm{ }}b} \right]\]
The cyclic arrangement of vectors in a scalar triple product affects the sign but not the magnitude of the product.
That is, \[\left[ {a{\rm{ }}b{\rm{ }}c} \right] = - \left[ {b{\rm{ }}a{\rm{ }}c} \right]{\rm{ = }} - \left[ {c{\rm{ }}b{\rm{ }}a} \right] = - \left[ {a{\rm{ }}c{\rm{ }}b} \right]\]
The placements of the dot and cross in a scalar triple product can be swapped as long as the vectors' cyclic order remains constant.
That is, \[\left( {a{\rm{ }} \times {\rm{ }}b} \right) \cdot c{\rm{ }} = {\rm{ }}a \cdot \left( {b{\rm{ }} \times {\rm{ }}c} \right)\]
If any two of the vectors are equal, the scalar triple product of the three vectors is zero.
Therefore, the term \[a \cdot (b \times c)\] is equal to \[b \cdot \left( {c \times a} \right)\]
Option ‘C’ is correct
Note: The relation \[\left[ {a{\rm{ }}b{\rm{ }}c} \right] = 0\]is a necessary and sufficient condition for three non-zero non-collinear vectors to be coplanar.
The expression\[\left[ {{\rm{ }}a{\rm{ }}b{\rm{ }}c} \right] + \left[ {d{\rm{ }}c{\rm{ }}a} \right] + \left[ {d{\rm{ }}a{\rm{ }}b} \right] = \left[ {a{\rm{ }}b{\rm{ }}c} \right]\], four points with position vectors ‘a’, ‘b’, ‘c’, and ‘d’ are coplanar.
The volume of a parallelepiped with coterminous edges a, b, and c is \[\left[ {a{\rm{ }}b{\rm{ }}c} \right]\] or \[a\left( {b{\rm{ }} \times {\rm{ }}c} \right)\].