Question

If \[a,b\] and \[c\] are three unit vectors such that \[a + b + c = 0\] where \[0\] is null vector, then \[a.b + b.c + c.a\] is equal to?
A.\[ - 3\]
B.\[ - 2\]
C.\[ - \dfrac{3}{2}\]
D.\[0\]

Answer 1

Hint: A vector is a quantity that has both magnitude, as well as direction. A vector that has a magnitude of \[1\] is a unit vector. It is also known as Direction Vector. Unit Vector is represented by the symbol \[' \wedge '\] which is called a cap or hat, such as \[\mathop a\limits^ \wedge \] . It is given by \[\mathop a\limits^ \wedge = \dfrac{a}{{\left| a \right|}}\] .
Where \[\left| a \right|\] is for the norm or magnitude of a vector \[a\] .

Complete step-by-step answer:
Some mathematical operations can be performed on vectors such as addition and multiplication. The multiplication of vectors can be done in two ways, i.e. dot product and cross product.
The definition of dot product can be given in two ways, i.e. algebraically and geometrically. Algebraically, the dot product is defined as the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the two vectors’ Euclidean magnitudes and the cosine of the angle between them. Both the definitions are equivalent when working with Cartesian coordinates. However, the dot product of two vectors is the product of the magnitude of the two vectors and the cos of the angle between them.
We can express the scalar product as:
\[a.b = \left| a \right|\left| b \right|\cos \theta \]
Where \[\left| a \right|\] and \[\left| b \right|\] represent the magnitude of the vectors \[a\] and \[b\] and \[\cos \theta \] denotes the cosine of the angle between both the vectors \[a\] and \[b\] and \[a.b\] indicate the dot product of the two vectors.
We know that \[{\left| {a + b + c} \right|^2} = \left( {a + b + c} \right).\left( {a + b + c} \right)\]
On simplifying this we get ,
\[{\left| {a + b + c} \right|^2} = {\left| a \right|^2} + {\left| b \right|^2} + {\left| c \right|^2} + 2\left( {a.b + b.c + c.a} \right)\]
Substituting the values we get ,
\[0 = 1 + 1 + 1 + 2\left( {a.b + b.c + c.a} \right)\]
Therefore we get ,
\[\left( {a.b + b.c + c.a} \right) = - \dfrac{3}{2}\]
Therefore option (C) is the correct answer.
So, the correct answer is “Option C”.

Note: Vector is a quantity that has both magnitude, as well as direction. A vector that has a magnitude of \[1\] is a unit vector. Dot product and cross product are two different things. Do the calculation part with great focus. Keep in mind that the magnitude of a unit vector is always \[1\] .