Classical Dynamics of Particles and Systems · Problem 1-19

Newtonian Mechanics—Single Particle — Marion & Thornton

Physics — Mechanics · vector-algebra-trigonometry

Problem

Derive the following expressions by using vector algebra: ((a)) cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta ((b)) sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta
Intuitive explanation

The key is to recognize that angles and trigonometric identities live naturally in the geometry of vectors. Construct two unit vectors at angles α\alpha and β\beta from the origin, then compute their dot product in two different ways: once using components (which gives you the trig identities) and once using the geometric definition a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta. The angle between them is α−β\alpha - \beta, so equating these two expressions yields both identities at once.

Intuitive explanation

The key is to recognize that angles and trigonometric identities live naturally in the geometry of vectors. Construct two unit vectors at angles α\alpha and β\beta from the origin, then compute their dot product in two different ways: once using components (which gives you the trig identities) and once using the geometric definition a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta. The angle between them is α−β\alpha - \beta, so equating these two expressions yields both identities at once.

Related theory

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