Trigonometric Dot Product The dot product is a natural function that can be defined on a pair of Euclidean vectors. Interestingly, it can be combined with the cosine law to provide insight into the angle (or that angle’s explementary angle) between two vectors. Thus, we can derive what can be called the trigonometric form of the dot

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Length of a Space Diagonal The Pythagorean theorem tells us how to compute the length of the hypotenuse of a right triangle in terms of its legs. The 3-dimensional analogue of this problem is to determine the distance between two opposite corners of a rectangular prism. Such a line segment is called a space diagonal and we show that

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Extended Law of Sines The law of sines is well-known, but its proof is not. It turns out that the expressions in the law of sines can be proven to all be equal to twice the circumradius, which not only proves the law of sines but extends it. We prove this result here, which says that $$frac{a}{sin A}=frac{b}{sin

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