‌ ‌If‌ ‌\[\bar‌ ‌A‌ ‌=‌ ‌2\hat‌ ‌i‌ ‌-‌ ‌3\hat‌ ‌j‌ ‌+‌ ‌7\hat‌ ‌k,\bar‌ ‌B‌ ‌=‌ ‌\hat‌ ‌i‌ ‌+‌ ‌2\hat‌ ‌j\]‌ ‌and‌ ‌\[\bar‌ ‌C‌ ‌=‌ ‌\hat‌ ‌j‌ ‌-‌ ‌
\hat‌ ‌k\].‌ ‌Then‌ ‌calculate‌ ‌\[\bar‌ ‌A.(\bar‌ ‌B‌ ‌\times‌ ‌\bar‌ ‌C)\].‌

Hint: Vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in three-dimensional space \[{R^3}\], and is denoted by the symbol = . Given two linearly independent vectors a and b, the cross product, a × b , is a vector that is perpendicular to both a and b, and thus normal to the plane containing them.

Complete step-by-step solution:
\[\bar A = 2\hat i - 3\hat j + 7\hat k,\bar B = \hat i + 2\hat j\] , \[\bar C = \hat j - \hat k\]
We are first calculating \[\bar B \times \bar C\] :by using a determinant method.
Therefore:
\[\bar B \times \bar C = \left( {\begin{array}{*{20}{c}}
  i&j&k \\
  1&2&0 \\
  0&1&{ - 2}
\end{array}} \right)\]
\[\bar B \times \bar C = \hat i(2 \times ( - 2) + 1 \times 0) - \hat j(1 \times ( - 2) + 0 \times 0) + \hat k(1 \times 1 + 0 \times 2)\]
\[\bar B \times \bar C = \hat i( - 4) - \hat j( - 2) + \hat k(1)\]
\[\bar B \times \bar C = - 4\hat i + 2\hat j + \hat k\]
Now , we have to perform dot product over the result of first result with \[\bar A\]
\[\bar A = 2\hat i - 3\hat j\]
\[\bar B \times \bar C = - 4\hat i + 2\hat j + \hat k\]
\[\bar A.(\bar B \times \bar C) = (2\hat i - 3\hat j).( - 4\hat i + 2\hat j + \hat k)\]
\[\bar A.(\bar B \times \bar C) = \hat i(2 \times ( - 4)) + \hat j( - 3 \times 2) + \hat k(0 \times 1)\]
\[\bar A.(\bar B \times \bar C) = - 8\hat i - 6\hat j\]

Note:-
Characteristics of the Vector product:
1. Vector product two vectors is always a vector.
2. Result of two vectors is perpendicular to a given plane.
3. The Vector product of two vectors is noncommutative.
4. Vector product obeys the distributive law of multiplication.