area of parallelogram in the plane with the cross productStewart Calculus 83 12.4.27Find the area of the parallelogram with vertices A(-3, 0), B(-1, 3), C(5, 2), and D(3, -1). These start at the far left and go around clockwise. Here is a plot.
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The area of the parallelogram is the product of their lengths times the sine of the included angle. So we calculate these and combine 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 cheating with the cross product (3 component vectors)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2Ji1GIzYqLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZYLUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GZW42JFEiM0YnRjlGaG4tRmVuNiRRIjBGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5RjkvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkhGV0ZZLUZQNiYtRiM2KkZURmBvRmhuRmNvRmhuRmNvRmZvRjlGOUZob0ZbcC1GNjYtUSI7RidGOUY7RltvRkBGQkZERkZGSEZcb0ZNLUYsNiVRI0FERidGL0YyRjUtRlA2Ji1GIzYqRmBvRmhuRlRGWkZobkZjb0Zmb0Y5RjlGaG9GW3BGXnBGYXBGZm9GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSgmdGltZXM7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkwtRiw2JVEjQURGJ0YvRjItRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4yNzc3Nzc4ZW1GJy1GLDYlUSVBcmVhRidGL0YyLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkgvRktGWUZYLUkmbXNxcnRHRiQ2Iy1GIzYmLUYsNiNRIUYnLUYjNictRiw2JVEiJUYnRi9GMi1GNjYtUSwmQ2VudGVyRG90O0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4xNjY2NjY3ZW1GJy9GTkZccEZlby8lK2V4ZWN1dGFibGVHRj1GOUZgb0Y5Rl5wRjk=