Question

‌ ‌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 ‌$\bar‌ ‌A.(\bar‌ ‌B‌ ‌\times‌ ‌\bar‌ ‌C)$.‌

Answer 1

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:

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.