Thornton & Marion, Classical Dynamics of Particles and Systems, 5th Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 07. Scalar Product of two (unit) vectors

The problem asks you to

1. Find the vectors describing the diagonals
2. The angle between diagonal vectors

This problem assumes

• A unit cube in the Cartesian (Rectangular) coordinate system

We should know about

1. Position vectors
2. Scalar Product of vectors

Solution

1) Find diagonal vectors

First, we can define the vector $latex \\vec{A}$ from the origin to (1,1,1). And, we can also define the vector $latex \\vec{B}$ from (0,0,1) to (1,1,0). Thus, a pair of diagonal vectors can be expressed as

$latex \\vec{A} = \\hat{i} + \\hat{j} + \\hat{k}$$latex \\vec{B} = \\hat{i} + \\hat{j} – \\hat{k}$2) The angle between diagonal vectors

From the scalar product,

$latex \\vec{A} \\cdot \\vec{B} = AB \\cos\\theta$ 

Since,

$latex \\vec{A} \\cdot \\vec{B} = (\\hat{i} + \\hat{j} + \\hat{k}) \\cdot (\\hat{i} + \\hat{j} – \\hat{k}) = 1 + 1 – 1 = 1$$latex AB \\cos\\theta = \\sqrt{3}\\sqrt{3} \\cos\\theta$ 

Therefore,

$latex \\theta = \\cos^{-1} (\\frac{1}{3}) = 70.53^\\circ$