Question

If a, b and c are unit coplanar vectors, then \[\left[ {2a - b\;2b - c\;2c - a} \right]\] is equal to
A. \[1\]
B. \[0\]
C. \[ - \sqrt 3 \]
D. \[\sqrt 3 \]
E. \[6\]

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.
Cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both vectors. The Vector product of two vectors, a and b, is denoted by \[a\times b\] . Its resultant vector is perpendicular to a and b. Vector products are also called cross products. Cross product of two vectors will give the resultant a vector and calculated using the Right-hand Rule.
We can express the vector product as:
\[a \times b = \left| a \right|\left| b \right|\sin \theta \]
Where \[\left| a \right|\] and \[\left| b \right|\] represent the magnitude of the vectors \[a\] and \[b\] and \[\sin \theta \] denotes the sine of the angle between both the vectors \[a\] and \[b\] and \[a \times b\] indicate the cross product of the two vectors.
If a, b and c are unit coplanar vectors, then \[\left[ {2a - b\;2b - c\;2c - a} \right]\] is equal to

Now, since a, b and c are unit coplanar vectors \[\left[ {abc} \right] = 0\]
 Because \[\left[ {abc} \right] = a.\left\{ {b \times c} \right\}\] and since b and c are unit coplanar vectors \[b \times c = 1\]
Therefore now \[\left[ {2a - b2b - c2c - a} \right]\]
\[ = \left( {2a - b} \right).\left\{ {\left( {2b - c} \right) \times \left( {2c - a} \right)} \right\}\]
On computing the cross product
\[ = \left( {2a - b} \right).\left\{ {4(b \times c) - 2(b \times a) + \left( {c \times a} \right)} \right\}\]
On simplification
\[ = 8\left( {a.(b \times c)} \right) - \left( {b.\left( {c \times a} \right)} \right)\]
On simplification
\[ = 8\left[ {abc} \right] - \left[ {abc} \right]\]
On subtracting
 \[= 7\left[ {abc} \right]\]
   \[= 0\]

So, the correct answer is “Option B”.

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 \[1\] .