Question

Show how the “Law of Cosines” derives from finding the magnitude of the difference of two vectors. As a consequence, show that if two vectors are perpendicular – then the law of cosines becomes the Pythagorean theorem.

Answer 1

The Law of Cosines is a general equation relating three sides and one angle in a triangle. There are no restrictions on the triangle's shape. Three elements determine a triangle. If any three of the four elements in the law-of-cosines equation are given, the equation allows you to calculate the fourth one. Figure A1 illustrates a general triangle. The three sides are labeled a, b, c, and the three angles are labeled α, β, γ. Figure A1

There are three law-of-cosines equations, depending on which angle is included:

c² = a² + b² - 2ab cos γ (A1)

a² = b² + c² - 2bc cos α (A2)

b² = c² + a² - 2ca cos β (A3)

Note that the Pythagorean theorem is a special case of these equations if one of the angles is equal to 90°. For example, if γ = 90°, then cos γ = 0 and Equation (A1) reduces to the Pythagorean theorem:

c² = a² + b² (A4)

Also note the minus sign in front of the cosine term in these equations. This has the following effect. Let's consider Equation (1). If γ < 90° , the cosine is positive. With the minus sign in front of the cosine term, Equation (A1) gives a value for c that is less than the value given by the Pythagorean theorem (4). If γ > 90°, the cosine is negative. Combined with the minus sign in front of the cosine term, the term now makes a positive contribution to the right-hand side of Equation (1) that yields a value of c that is greater than the one given by the Pythagorean theorem.