if \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) and \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) are reciprocal system of vectors then \(a \times p+b \times q+c \times r\) is equal to (a) \([\mathrm{a} \mathrm{b} \mathrm{c}]\) (b) \(3(a+b+c)\) (c) 0 (d) None of these.

To find the value of \)a \times p+b \times q+c \times r$, we need to perform vector cross products for each term and then add them together.

Now, let's expand the three cross products. \[a \times p = a_1 i + a_2 j + a_3 k\] \[b \times q = b_1 i + b_2 j + b_3 k\] \[c \times r = c_1 i + c_2 j + c_3 k\]

Now, let's add all the expanded cross products to get our final result. \[(a_1 i + a_2 j + a_3 k) + (b_1 i + b_2 j + b_3 k) + (c_1 i + c_2 j + c_3 k)\] Combine the terms corresponding to \(i\), \(j\), and \(k\): \[(a_1 + b_1 + c_1)i + (a_2 + b_2 + c_2)j + (a_3 + b_3 + c_3)k\] Now, compare this expression with the given options: (a) \([\mathrm{a} \mathrm{b} \mathrm{c}]\) This represents the scalar triple product, which is not equal to the given expression. (b) \(3(a+b+c)\) This represents a scalar multiple of the sum of vectors a, b, and c. This is not equal to the given expression. (c) 0 Option (c) suggests that the expression is equal to the zero vector. To be equal to the zero vector, all the coefficients of \(i, j,\) and \(k\) must be equal to zero: \[a_1 + b_1 + c_1 = 0\] \[a_2 + b_2 + c_2 = 0\] \[a_3 + b_3 + c_3 = 0\] These conditions are not necessarily true for the reciprocal system of vectors. Therefore, the expression is not always equal to zero. (d) None of these Since none of the above three options is equal to the given expression, the correct answer is (d) None of these.