
\[a \cdot (b \times c)\] is equal to
A. \[b \cdot (a \times c)\]
B. \[c \cdot (b \times a)\]
C. It is obvious.
D. None of these
Answer
160.8k+ views
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 \[a\left( {b{\rm{ }} \times {\rm{ }}c} \right)\].
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

JEE Advanced 2025 Notes

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

List of Fastest Century in IPL History
