Proving the Law of Cosines Using Dot Product Vector Algebra

The Law of Cosines is a fundamental principle in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. However, it can also be derived using the properties of dot product in vector algebra. In this article, we will explore how the Law of Cosines can be proved using the dot product of vectors.

Introduction to the Dot Product

The dot product of two vectors u and v is defined as:

[ u cdot v u_x v_x u_y v_y ]

This can also be expressed in terms of the magnitudes of the vectors and the cosine of the angle between them:

[ u cdot v |u||v|cos C ]

The Derivation of the Law of Cosines

Consider a triangle ABC where the vectors u and v are drawn from the origin C00 to points A C v (v_x, v_y) and B C u (u_x, u_y) respectively. Let a, b, and c be the lengths of the sides opposite to vertices A, B, and C respectively.

We are interested in expressing c in terms of a, b, and the angle C between the vectors u and v. The vectors representing the sides are:

a BC u b AC v c AB v - u

Starting from the dot product formula for vectors u and v: [ u cdot v u_x v_x u_y v_y |u||v|cos C ]

We can manipulate this equation to derive the Law of Cosines:

[ u_x v_x u_y v_y |u||v|cos C ]

Multiplying both sides by -2, we get:

[ -2(u_x v_x u_y v_y) -2|u||v|cos C ]

Now, let's add the squares of the magnitudes of u and v to both sides:

[ u_x^2 u_y^2 v_x^2 v_y^2 - 2(u_x v_x u_y v_y) |u|^2 |v|^2 - 2|u||v|cos C ]

This simplifies to:

[ u_x^2 - 2u_x v_x v_x^2 u_y^2 - 2u_y v_y v_y^2 a^2 b^2 - 2abcos C ]

Which further simplifies to:

[ (v_x - u_x)^2 (v_y - u_y)^2 a^2 b^2 - 2abcos C ]

This can be written as:

[ c^2 a^2 b^2 - 2abcos C ]

Alternative Derivation Using Vectors and Modulus

Consider three vectors such that:

[ vec{a} - vec{b} vec{c} ]

Taking the modulus on both sides of this equation gives us:

[ |vec{a} - vec{b}| |vec{c}| ]

Squaring both sides, we get:

[ |vec{a} - vec{b}|^2 |vec{c}|^2 ]

The modulus squared of the difference of two vectors can be expanded using the dot product:

[ |vec{a} - vec{b}|^2 (vec{a} - vec{b}) cdot (vec{a} - vec{b}) ]

This expands to:

[ |vec{a}|^2 - 2vec{a} cdot vec{b} |vec{b}|^2 |vec{c}|^2 ]

Which simplifies to:

[ a^2 - 2vec{a} cdot vec{b} b^2 c^2 ]

Since the angle between a and b is actually 180° - C, we can write:

[ a^2 b^2 - 2abcos(180° - C) c^2 ]

Since (cos(180° - C) -cos C), we get:

[ a^2 b^2 - 2ab(-cos C) c^2 ]

Simplifying further:

[ a^2 b^2 2abcos C c^2 ]

Or, more simply:

[ c^2 a^2 b^2 - 2abcos C ]

Conclusion

The Law of Cosines is a powerful tool in trigonometry that can be derived using the properties of dot product in vector algebra. By manipulating the dot product and using the properties of vector addition and subtraction, we have shown that the Law of Cosines is indeed a valid and useful formula for any triangle.