a is perpendicular with \(\underline{\mathrm{b}}\) and \(\underline{\mathrm{c}}\) then (a) \(\underline{\mathrm{a}} \times(\underline{\mathrm{b}} \times \underline{\mathrm{c}})=1\) (b) \(\underline{\mathrm{a}} \times(\underline{\mathrm{b}} \times \underline{\mathrm{c}})=0\) (c) \(\underline{\mathrm{a}} \times(\underline{\mathrm{b}} \times \underline{\mathrm{c}})=-1\) (d) none of these.

We need to calculate the cross product of vector \(\underline{\mathrm{a}}\) with the cross product of vectors \(\underline{\mathrm{b}}\) and \(\underline{\mathrm{c}}\). To do this, we should recall some important properties of the cross product: 1. The cross product of two vectors is a vector that is orthogonal (perpendicular) to both of the original vectors. 2. The cross product is distributive, i.e., \( \underline{\mathrm{a}} \times (\underline{\mathrm{b}} + \underline{\mathrm{c}}) = \underline{\mathrm{a}} \times \underline{\mathrm{b}} + \underline{\mathrm{a}} \times \underline{\mathrm{c}} \). 3. The cross product is anti-commutative, i.e., \( \underline{\mathrm{a}} \times \underline{\mathrm{b}} = -(\underline{\mathrm{b}} \times \underline{\mathrm{a}}) \).

Let's denote the cross product of vectors \(\underline{\mathrm{b}}\) and \(\underline{\mathrm{c}}\) as vector \(\underline{\mathrm{d}}\). From property 1, we know that vector \(\underline{\mathrm{d}}\) is orthogonal to both \(\underline{\mathrm{b}}\) and \(\underline{\mathrm{c}}\): \[ \underline{\mathrm{d}} = \underline{\mathrm{b}} \times \underline{\mathrm{c}} \]

Now, we want to calculate the cross product of vector \(\underline{\mathrm{a}}\) with vector \(\underline{\mathrm{d}}\). Since \(\underline{\mathrm{d}}\) is orthogonal to both \(\underline{\mathrm{b}}\) and \(\underline{\mathrm{c}}\), and \(\underline{\mathrm{a}}\) is also orthogonal to both \(\underline{\mathrm{b}}\) and \(\underline{\mathrm{c}}\), we can conclude that \(\underline{\mathrm{a}}\) and \(\underline{\mathrm{d}}\) are either parallel or antiparallel. Therefore, their cross product will be a zero vector: \[ \underline{\mathrm{a}} \times (\underline{\mathrm{b}} \times \underline{\mathrm{c}}) = \underline{\mathrm{a}} \times \underline{\mathrm{d}} = \underline{\mathrm{0}} \]

The cross product of vector \(\underline{\mathrm{a}}\) with the cross product of vectors \(\underline{\mathrm{b}}\) and \(\underline{\mathrm{c}}\) is the zero vector. Hence, the correct option is: (b) \(\underline{\mathrm{a}} \times (\underline{\mathrm{b}} \times \underline{\mathrm{c}}) = 0\)