z1=(1+i)/sqrt(2),z2=(sqrt(3)+i)/2, Q1:|z1+z2|, Q2:cos(7.5°)

sympy.ipynb

{"cells":[{"cell_type":"code","execution_count":1,"metadata":{},"outputs":[],"source":["from sympy import *\n","from sympy.physics.vector import *\n","\n","init_printing(fold_short_frac=False)\n",""]},{"cell_type":"code","execution_count":2,"metadata":{},"outputs":[],"source":["N= ReferenceFrame('N')\n",""]},{"cell_type":"code","execution_count":3,"metadata":{},"outputs":[{"data":{"image/png":"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\n","text/latex":"$$\\frac{\\sqrt{2}}{2}\\mathbf{\\hat{n}_x} + \\frac{\\sqrt{2}}{2}\\mathbf{\\hat{n}_y}$$","text/plain":"√2       √2\n── n_x + ── n_y\n2        2"},"execution_count":3,"metadata":{},"output_type":"execute_result"}],"source":["u= 1/sqrt(2) * N.x + 1/sqrt(2) * N.y;u\n",""]},{"cell_type":"code","execution_count":4,"metadata":{},"outputs":[{"data":{"image/png":"iVBORw0KGgoAAAANSUhEUgAAAG0AAAAkCAYAAACDr7TyAAAABHNCSVQICAgIfAhkiAAABAVJREFUaIHt2nvIX3McwPHXzGzyKLY/DKm1JJbVFiGynpZSxCJhc3sSc01Sir/IP4TV3Cax9ihZVi6R/EFRNLc/DFFCDWsyehYthvHzx+ecfpf9bs+5/fbkvOt0fs/3e87n9r2cz/mch5oZx6whrmmULL+mBG4dtQE17Rw0oH82/q7CkJriWIGlozaipp1BK20pPq/CkJrhGTRonUnILfgMvyXH+zi/BLsONFbgNewUMblklMYc3KfvBHzV0bYDd+FrkRleg1dwihjM6bIcq5LfW/BlBhlVcBg+xSa8mENO4f5erX3VXI85Q9w3hRsy6Dsc3+JfMXu/wKEZ5FRN1pVWmL/p9jgXu7CmpW+O/pnjbKzGGLZm0P0EFuN2sXqX4OEMcmYKpfn7ntgyF+DSHtcsxR7sw26cl0HP5WK23dPS9mDSdkEGeVWSZaWV6u/9WJkoOaLHNYfgeJyaXP8LTs6reAYx8kSkkzPwCG6axj1v4ZlyzCmUCRHw8ZxyRj5onSn/h2Lg/pymjHl9+ieEo+mxGDdiG/4QK/V5HDMNnQcyE0r2tzPlb4jU/Z0e1z+A1/GDyIbWiJk7nXe1SZzd8vc8kdAsxzLTmzBVMSYeCSmLhK1T+H7AvZMq8HfBAAO+SxTtElvjuQPkTWifeY3kvvvEBGltvyy72QNJ7RjPcO+4/X1oiHj00jNqf3Mxod3QlzQ/2cwXWWjat64CO8ZL1NGqpzR/B5WxyuBJzfLYlNjjU44sSMd2+8/2TUnf2136JgvS243C/e1XxiqL7R1/t+7pRU2i9fZ/ZVkmSkjPdrFhW0F6u9GpK7e/oxi0zipLni/jvVjfpW1CDNqk3olWGRTubzrS3R6yRRx5uLBFzjlJ20rN2t1FOeUPS1X+bk5kfNDRvjVp35w2pIM2q6QjD6/iqeT301iIjYncjXg5p/xhqcrfDcn5dFGXhGPFezOxrWO4PfVufCy+n/0svitVVba6Q3weWiSeO4vwjSi6VkkVMXhX84Pztcn5YjEZduLN9MJhBm1czIIzxfa0T7x3zC/G1r78jivwD44SW+OVomBdJeOqiUG62q4S+UZaLntOxCAzY4mAqqrxq7Q/N1ZXpLcfZcVgDL8KP9cmOho4Ka/goxNBZ+UVNAQLxXbUwCfJeTeOq0B3P8qMweOJ7D3J+aMihL4gAji7CGF9mIU3NA2fKwraDfGCPIrCQEqZMViifWe5Oa/Ah/Cj9uJpWdwmjN6ruT2cKCrlDdxZgQ3dqCIGadVmr5zPzXX4STMd/T9SVQweE4O2JY+QR9UDVkUM1or/bvtLZMqnZRW0QbyfrBSJQXqM5bdxxlBVDCbFCtuB6/II6lWyuTeXeTOLOgY1NTU1NTUl8B/FLEvoJzjTAQAAAABJRU5ErkJggg==\n","text/latex":"$$\\frac{\\sqrt{3}}{2}\\mathbf{\\hat{n}_x} + \\frac{1}{2}\\mathbf{\\hat{n}_y}$$","text/plain":"√3\n── n_x + 1/2 n_y\n2"},"execution_count":4,"metadata":{},"output_type":"execute_result"}],"source":["v= sqrt(3)/2 * N.x + Rational(1, 2) * N.y;v\n",""]},{"cell_type":"code","execution_count":5,"metadata":{},"outputs":[{"data":{"image/png":"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\n","text/latex":"$$(\\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{3}}{2})\\mathbf{\\hat{n}_x} + (\\frac{1}{2} + \\frac{\\sqrt{2}}{2})\\mathbf{\\hat{n}_y}$$","text/plain":"⎛√2   √3⎞       ⎛1   √2⎞\n⎜── + ──⎟ n_x + ⎜─ + ──⎟ n_y\n⎝2    2 ⎠       ⎝2   2 ⎠"},"execution_count":5,"metadata":{},"output_type":"execute_result"}],"source":["w= u+v;w\n",""]},{"cell_type":"code","execution_count":6,"metadata":{},"outputs":[{"data":{"image/png":"iVBORw0KGgoAAAANSUhEUgAAAHkAAAAqCAYAAACX4PQQAAAABHNCSVQICAgIfAhkiAAAA6dJREFUeJzt20uIHEUYwPFfTEIQNxqyF8WDEERFsgcFNSii5OIjvk4ioqJiDj4i4k1FCAoqiK8gexAP6yUYUMQoGkVkQdRIDj4RFERy8skSxSSarK6HbwaHdrqne3p6e3ZTfygYqqu6/tNfV/fXXTMklj0rWh7/RPzWskOiYe7GVNsSy53jWh5/I75s2WHZ02aQJ/B7i+MfM7QZ5MvxbovjJxaBHVjdtsSxQJmZvFCj5LFSZPZHS3o24VCXC8bQqS9lgrxNBGSYkseF+KiCZxMOdTlvDJ36MijIK5WfbVW4Am+XbNuUQx3G0SmXQUG+SLUZV5Z1OFCybVMOdRhHp1wGBXlKvefYbTg3U3cGvqmwj7oOTTCOTrkMCnI2SXgA+8Tz7S94Q7zQ6MdVuAPXZ+qvxu4KjnUcmiIveToFLwmvw+JEuGQRfB7sOD3fb2NRkPvNuEsxLRKnzZjHe1jfp/+buB/XZuo34PsB0qNyKGIG2yv2yXMibkEfisRqC87Gffi5YadN2IovyjS+Rch12Wrwc+wE/hazsx+rMCcODEzikUV2yGNGuQNa1ukxEeQ6lHXqchK+Eyf7rAEzeY04427s2bba4AxybWcfcznb50UW3Z3NV+KtnLZNOdShitN12IudnT6f4R7NPjK9gFfwflGjbpD/wh6cJmbfJH4tMciz4svsLWjzOq7pfD4fn+S0a9JhWKo4bRCravtxGZ7DE7irAS/iinI6Hq7a8XEx9W8Q95ginsQPnYGKWItDOBVPt+RAJCd/9JSjOJKpu7iG0xF83Kff1w04nSmSu7N66mblXK6zbBJn4J0D2j2Fn0RyUYY9eNl/M7oNh/XiZOiWV8X7896642s47ceLmbqbcbABp1tFNj3fUxbwT+fzmoIxrRCX09sL2uxQ7eASl6xDOcKL5ZBlRvkkp4zTTnyQqXtU8Uwe1mmdeGzsLfs6Dhtl8oBVmc4LIhWfzdn5NG4SScYcTu7Udy8teewW96nDJb5AUw51GOQEz4i3YA9hF87BveKSPGoO+P8bw4PieHxVZgeTBdvyVlW2l9jvCWUGb9ihl5mKfYqcumzB5/gT34ogV8muqzr1MqvkPTmRSCQSiUQ7dJOCRf9JSiKRSFRmHNagEw3zDm4TgZ3Ca/hR9TXoxBJi2DXoJUnb/4VqiybXoBNjwi58Kn5am1iGVFmDTixBqq5BJ5YYo1iDTowx0+IZebNYf+6WiTalEqNlVGvQiUQikUgkEkPzLwytJZL6ZXGXAAAAAElFTkSuQmCC\n","text/latex":"$$\\frac{\\sqrt{2} \\sqrt{\\sqrt{2} + \\sqrt{6} + 4}}{2}$$","text/plain":"     _____________\n√2⋅╲╱ √2 + √6 + 4 \n──────────────────\n        2         "},"execution_count":6,"metadata":{},"output_type":"execute_result"}],"source":["a= factor(w.magnitude());a\n",""]},{"cell_type":"code","execution_count":7,"metadata":{},"outputs":[{"data":{"image/png":"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\n","text/latex":"$$\\frac{\\sqrt{2} \\sqrt{\\sqrt{2} + \\sqrt{6} + 4}}{4}$$","text/plain":"     _____________\n√2⋅╲╱ √2 + √6 + 4 \n──────────────────\n        4         "},"execution_count":7,"metadata":{},"output_type":"execute_result"}],"source":["c= factor((v & w) / (v.magnitude() * w.magnitude()));c\n",""]},{"cell_type":"code","execution_count":8,"metadata":{},"outputs":[{"data":{"image/png":"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\n","text/latex":"$$7.5$$","text/plain":"7.50000000000000"},"execution_count":8,"metadata":{},"output_type":"execute_result"}],"source":["(acos(c)/pi*180).evalf()\n",""]}],"nbformat":4,"nbformat_minor":2,"metadata":{"language_info":{"name":"python","codemirror_mode":{"name":"ipython","version":3}},"orig_nbformat":2,"file_extension":".py","mimetype":"text/x-python","name":"python","npconvert_exporter":"python","pygments_lexer":"ipython3","version":3}}