# 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

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,

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