Question

If \[a\] is perpendicular to \[b\] and \[c\], \[\left| a \right| = 2\], \[\left| b \right| = 3\], \[\left| c \right| = 4\]and the angle between \[b\] and \[c\]is \[\dfrac{{2\pi }}{3}\]. What is the value of \[\left[ {a\,b\,c} \right]\]?
A. \[4\sqrt 3 \]
B. \[6\sqrt 3 \]
C. \[12\sqrt 3 \]
D. \[18\sqrt 3 \]

Answer 1

Hint The formula of \[\left[ {a\,b\,c} \right]\] is \[a \cdot \left( {b \times c} \right)\]. To calculate \[\left[ {a\,b\,c} \right]\], first we will find dot product between \[\left( {b \times c} \right)\] and \[a\] by using the formula \[\left| a \right| \cdot \left| {b \times c} \right|\cos \theta \]. Then apply the formula of cross product \[\left| b \right|\left| c \right|\sin \theta \] on \[\left( {b \times c} \right)\] and find the magnitude of the vector \[\left| b \right|\left| c \right|\sin \theta \]. Then multiply the magnitude of \[\left| {b \times c} \right|\]with \[\left| a \right|\].

Formula used
\[a \cdot b = \left| a \right|\left| b \right|\cos \theta \], where \[\theta \] is angle between \[a\] and \[b\].
\[\left| {a \times b} \right| = \left| a \right|\left| b \right|\sin \theta \], where \[\theta \] is angle between \[a\] and \[b\].
\[\left[ {a\,b\,c} \right] = a \cdot \left( {b \times c} \right)\]

Complete step by step solution
Given triple product is \[\left[ {a\,b\,c} \right]\].
Apply the formula \[\left[ {a\,b\,c} \right] = a \cdot \left( {b \times c} \right)\]
\[\left[ {a\,b\,c} \right] = a \cdot \left( {b \times c} \right)\]
\[a\] is perpendicular to \[b\] and \[c\]and the cross product of \[b\] and \[c\] is a vector that is perpendicular to the plane where \[b\] and \[c\] lie. So, \[a\] and \[\left( {b \times c} \right)\] is parallel to each other. Therefore the angle between \[a\] and \[\left( {b \times c} \right)\] is \[{0^ \circ }\].
Apply dot product formula \[a \cdot b = \left| a \right|\left| b \right|\cos \theta \] on \[a \cdot \left( {b \times c} \right)\]
\[\left[ {a\,b\,c} \right] = \left| a \right|\left| {b \times c} \right|\cos {0^ \circ }\]
Now putting \[\cos {0^ \circ } = 1\].
\[\left[ {a\,b\,c} \right] = \left| a \right|\left| {b \times c} \right|\] …….(1)
Now apply cross product formula on \[\left( {b \times c} \right)\]
\[\left| {b \times c} \right| = \left| b \right|\left| c \right|\sin \dfrac{{2\pi }}{3}\] [Since the angle between \[b\] and \[c\]is \[\dfrac{{2\pi }}{3}\]]
Now we will calculate the value of \[\sin \dfrac{{2\pi }}{3}\].
\[\sin \dfrac{{2\pi }}{3} = \sin \left( {\dfrac{\pi }{2} + \dfrac{\pi }{6}} \right)\]
Apply complement formula \[\sin \left( {\dfrac{\pi }{2} + \theta } \right) = \cos \theta \]
\[\sin \dfrac{{2\pi }}{3} = \cos \left( {\dfrac{\pi }{6}} \right)\]
\[ \Rightarrow \sin \dfrac{{2\pi }}{3} = \dfrac{{\sqrt 3 }}{2}\]
Now we will put the value of \[\left| b \right|\], \[\left| c \right|\] and \[\sin \dfrac{{2\pi }}{3}\] in \[\left| {b \times c} \right| = \left| b \right|\left| c \right|\sin \dfrac{{2\pi }}{3}\]
\[\left| {b \times c} \right| = 3 \cdot 4 \cdot \dfrac{{\sqrt 3 }}{2}\]
\[ \Rightarrow \left| {b \times c} \right| = 6\sqrt 3 \]
Now we will calculate the value of \[\left[ {a\,b\,c} \right] = \left| a \right|\left| {b \times c} \right|\] by substituting the value of \[\left| a \right|\] and \[\left| {b \times c} \right|\].
\[\left[ {a\,b\,c} \right] = 2 \cdot 6\sqrt 3 \]
\[ \Rightarrow \left[ {a\,b\,c} \right] = 12\sqrt 3 \]
Hence the correct option is C.

Note Students are confused with the formula cross product and magnitude of a cross product. Magnitude of a cross product is a scalar and cross product is a vector. The formula of cross product is \[a \times b = \left| a \right|\left| b \right|\sin \theta \widehat n\] and the magnitude of a cross product is \[\left| {a \times b} \right| = \left| a \right|\left| b \right|\sin \theta \]. Here we need to apply the formula \[\left| {a \times b} \right| = \left| a \right|\left| b \right|\sin \theta \] to get the value of \[\left| {b \times c} \right|\].