Question

If \[\vec a,\vec b,\vec c\] are vectors such that \[\left[ {\vec a\vec b\vec c} \right] = 4\] then find the value of \[\left[ {\begin{array}{*{20}{c}}{\vec a \times \vec b}&{\vec b \times \vec c}&{\vec c \times \vec a}\end{array}} \right]\]
A. \[14\]
B. \[15\]
C. \[16\]
D. \[5\]

Answer 1

Hint: In the given question, we need to find the value of \[\left[ {\begin{array}{*{20}{c}}{\vec a \times \vec b}&{\vec b \times \vec c}&{\vec c \times \vec a}\end{array}} \right]\]. For this, we will use the property of box product of vectors to get the desired result.

Formula used: The following formula used for solving the given question.
The property of box product is given by
\[\left[ {\begin{array}{*{20}{c}}{\vec a \times \vec b}&{\vec b \times \vec c}&{\vec c \times \vec a}\end{array}} \right] = {\left[ {\vec a\vec b\vec c} \right]^2}\]
Here, \[\vec a,\vec b,\vec c\] are vectors.

Complete step by step solution: We know that \[\left[ {\vec a\vec b\vec c} \right] = 4\]
Here, we will find the value of \[\left[ {\begin{array}{*{20}{c}}{\vec a \times \vec b}&{\vec b \times \vec c}&{\vec c \times \vec a}\end{array}} \right]\].
According to property of box product of vectors, we get
\[\left[ {\begin{array}{*{20}{c}}{\vec a \times \vec b}&{\vec b \times \vec c}&{\vec c \times \vec a}\end{array}} \right] = {\left[ {\vec a\vec b\vec c} \right]^2}\]
Thus, we can say that
\[\left[ {\begin{array}{*{20}{c}}{\vec a \times \vec b}&{\vec b \times \vec c}&{\vec c \times \vec a}\end{array}} \right] = {\left( 4 \right)^2}\]
By simplifying, we get
\[\left[ {\begin{array}{*{20}{c}}{\vec a \times \vec b}&{\vec b \times \vec c}&{\vec c \times \vec a}\end{array}} \right] = 16\]
Hence, the value of \[\left[ {\begin{array}{*{20}{c}}{\vec a \times \vec b}&{\vec b \times \vec c}&{\vec c \times \vec a}\end{array}} \right]\] is \[16\] if If \[\vec a,\vec b,\vec c\] are vectors such that \[\left[ {\vec a\vec b\vec c} \right] = 4\].

Thus, Option (C) is correct.

Additional Information:The definition of a vector is an entity with both magnitude and direction. The movement of an object between two points is described by a vector. The directed line segment can be used to geometrically represent vector mathematics. The magnitude of a vector is the length of the directed line segment, and the vector's direction is indicated by the angle at which it is inclined. A vector's "Tail" (the point where it begins) and "Head" (the point where it ends and has an arrow) are its respective names.

Note: Many students make mistake in writing the property of box product of vectors. This is the only way, through which we can solve the example in simplest way. Also, it is necessary to use the fact such as \[\left[ {\vec a\vec b\vec c} \right] = 4\] to get the desired result.