Thornton & Marion (5th Edition), Chapter 01, Exercise Problem 07

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 \vec{A} from the origin to (1,1,1).

And, we can also define the vector \vec{B} from (0,0,1) to (1,1,0).

Thus, a pair of diagonal vectors can be expressed as

\vec{A} = \hat{i} + \hat{j} + \hat{k}
\vec{B} = \hat{i} + \hat{j} - \hat{k}
2) The angle between diagonal vectors From the scalar product,
\vec{A} \cdot \vec{B} = AB \cos\theta
Since,
\vec{A} \cdot \vec{B} = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} + \hat{j} - \hat{k}) = 1 + 1 - 1 = 1
AB \cos\theta = \sqrt{3}\sqrt{3} \cos\theta
Therefore,
\theta = \cos^{-1} (\frac{1}{3}) = 70.53^\circ

 

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