Let's compute the cross product step by step:\n\nThe cross product (mathbf{B} = mathbf{T} \times mathbf{N}) is given by the determinant:\n\n[\nmathbf{B} = \begin{vmatrix}\nmathbf{i} & mathbf{j} & mathbf{k} \\n-4 & 0 & 16 \\n0 & -16 & 0\nend{vmatrix}\n]\n\nExpanding this determinant:\n\n[\nmathbf{B} = mathbf{i} left( 0 cdot 0 - 16 cdot (-16) \right) - mathbf{j} left( -4 cdot 0 - 16 cdot 0 \right) + mathbf{k} left( -4 cdot (-16) - 0 cdot 0 \right)\n]\n\nSimplifying each component:\n\nFor the (mathbf{i}) component:\n[\nmathbf{i} (0 - (-256)) = mathbf{i} cdot 256\n]\n\nFor the (mathbf{j}) component:\n[\n- mathbf{j} (0 - 0) = 0\n]\n\nFor the (mathbf{k}) component:\n[\nmathbf{k} (64 - 0) = mathbf{k} cdot 64\n]\n\nTherefore, the cross product is:\n\n[\nmathbf{B} = 256mathbf{i} + 0mathbf{j} + 64mathbf{k}\n]\n\nSo,\n\n[\nmathbf{B} = 256mathbf{i} + 64mathbf{k}\n]\n\nWould you like any further details or have any questions about this computation?\n\nHere are 5 questions you might consider next:\n1. How do you find the magnitude of a vector?\n2. What are some applications of the cross product in physics?\n3. How do you determine if two vectors are perpendicular?\n4. How can you express a vector in polar coordinates?\n5. What is the dot product and how is it different from the cross product?\n\nTip: When computing determinants, always remember to carefully handle signs and zeros to avoid common errors.