Your task in 3D: For each of the transformations listed below, sketch a picture of the transformation to decide if you think the transformation preserves length and/or orientation (again, orientation is hard to intuit here, take your best guess). Compute the determinant of the matrix that performs that transformation. Summarize your findings and make a conjecture about how the determinant relates to length and orientation preservation. 1. Dilate each vector by a factor of 5 2. Rotate each vector an angle of theta about the x_3 axis counterclockwise (when viewed from the +x_3 axis) and then by an angle of phi about the x_2 axis counterclockwise (when viewed from the +x_2 axis). 3. Reflect each vector across the x1x2 plane. (hint: a good way to construct this matrix is to think about what happens to the basis vectors) 4. Reflect each vector across the x1x2 plane, then across the x2x3 plane. Your task in 3D: For each of the transformations listed below, sketch a picture of the transformation to decide if you think the transformation preserves length and/or orientation (again, orientation is hard to intuit here, take your best guess). Compute the determinant of the matrix that performs that transformation. Summarize your findings and make a conjecture about how the determinant relates to length and orientation preservation. 1. Dilate each vector by a factor of 5 2. Rotate each vector an angle of theta about the x_3 axis counterclockwise (when viewed from the +x_3 axis) and then by an angle of phi about the x_2 axis counterclockwise (when viewed from the +x_2 axis). 3. Reflect each vector across the x1x2 plane. (hint: a good way to construct this matrix is to think about what happens to the basis vectors) 4. Reflect each vector across the x1x2 plane, then across the x2x3 plane