Question

If\[a,b,c\]are mutually perpendicular vectors then \[[abc] = \]
A) 0
B) 1
C) 3
D) \[ \pm abc\]

Answer 1

Hint:
Here, we will use the properties of the vectors to find the value of the given expression. Firstly by the identity of the vectors we will equate the given expression. Then we will substitute the values in the expression to get the value of the expression.

Complete step by step solution:
We know that if two vectors are perpendicular to each other, then the dot product of the vectors is always zero. Therefore,
\[a \cdot b = \left| a \right|\left| b \right|\cos \theta = 0\]
If the dot product between the two perpendicular vectors is zero, then the value of the angle between the vectors is \[90^\circ \]. Therefore,
\[a \cdot b = 0\]
\[b \cdot c = 0\]
\[a \cdot c = 0\]
Now we will expand the given expression \[[abc]\]. Expanding the expression, we get
\[ \Rightarrow [abc] = a \cdot (b \times c)\]
Using the distributive property to simplify the expression, we get
\[ \Rightarrow [abc] = a \cdot b \times a \cdot c\]
Substituting the value \[a \cdot b = 0\] and \[a \cdot c = 0\] in the above equation, we get
\[ \Rightarrow [abc] = 0 \times 0\]
\[ \Rightarrow [abc] = 0\]
Hence, 0 is the value of the given expression \[[abc]\].

So, option A is the correct option.

Note:
Here we have to keep in mind that while expanding the dot product of vectors is related to cos function not the sin function. Sin function is related to the cross product of the vectors. Also we have to remember that the dot product of the perpendicular vectors is always zero as the angle between them is \[90^\circ \]. Also the cross product of the parallel vectors is always zero as the angle between the parallel vectors is \[0^\circ \] or \[{\rm{180}}^\circ \].
Dot product of two vectors is given by \[a \cdot b = \left| a \right|\left| b \right|\cos \theta \].
Cross product of two vectors is given by \[a \times b = \left| a \right|\left| b \right|\sin \theta \].
Here, \[\theta \] is the angle between the vectors.