Question

If \[a = 2i + 3j - k,\,b = - i + 2j - 4k\] and \[c = i + j + k\], then what is the value of \[\left( {a \times b} \right).\left( {a \times c} \right)\]
a) 47
b) 74
c) -74
d) None of these.

Answer 1

Hint: Here in this question we have to determine the value of \[\left( {a \times b} \right).\left( {a \times c} \right)\]. First we determine the dot products and then we determine the cross product. Since the vectors contain the unit vectors i, j and k, the properties are implemented to it and then on simplification we get a desired result.

Complete step-by-step answer:
Here we have 3 vectors \[a = 2i + 3j - k,\,b = - i + 2j - 4k\] and \[c = i + j + k\].
Now we have to determine \[\left( {a \times b} \right).\left( {a \times c} \right)\]
The symbol \[ \times \] represents the cross product and \[.\] represents the dot product.
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 \[\left( {a \times b} \right)\]. Its resultant vector is perpendicular to a and b. Vector products are also called cross products.
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
Now first we determine \[\left( {a \times b} \right).\left( {a \times c} \right) = \left| {\begin{array}{*{20}{c}}
  {a.a}&{a.c} \\
  {b.a}&{b.c}
\end{array}} \right|\] --- (1)
The dot product of a vector and a vector is
\[ \Rightarrow a.a = (2)(2) + (3)(3) + ( - 1)( - 1)\]
On multiplying we get
\[ \Rightarrow a.a = 4 + 9 + 1 = 14\].
The dot product of a vector and c vector is
\[ \Rightarrow a.c = (2)(1) + (3)(1) + ( - 1)(1)\]
On multiplying we get
\[ \Rightarrow a.a = 2 + 3 - 1 = 4\]
The dot product of b vector and a vector is
\[ \Rightarrow b.a = ( - 1)(2) + (2)(3) + ( - 4)( - 1)\]
On multiplying we get
\[ \Rightarrow b.a = - 2 + 6 + 4 = 8\].
.The dot product of b vector and c vector is
\[ \Rightarrow b.c = ( - 1)(1) + (2)(1) + ( - 4)(1)\]
On multiplying we get
\[ \Rightarrow b.c = - 1 + 2 - 4 = - 3\].
On substituting the above values in the equation (1) we get
\[ \Rightarrow \left( {a \times b} \right).\left( {a \times c} \right) = \left| {\begin{array}{*{20}{c}}
  {14}&4 \\
  8&{ - 3}
\end{array}} \right|\]
On simplification
\[ \Rightarrow \left( {a \times b} \right).\left( {a \times c} \right) = - 42 - 32\]
\[ \Rightarrow \left( {a \times b} \right).\left( {a \times c} \right) = - 74\]
Hence the option c) is the correct one.
So, the correct answer is “Option C”.

Note: The dot product of the unit vectors are given as \[i.i = 1,j.j = 1\], \[k.k = 1\] and \[i.j = j.k = k.i = 0\]. The cross product of unit vectors are given as \[i \times j = k,\,j \times k = i,k \times i = j\] and \[i \times i = j \times j = k \times k = 0\]. While determining the value for the determinant, the sign conventions are important.