Preliminary Knowledge

Abstract

In this chapter, we give some fundamental knowledge to learn Theoretical Mechanics, which is very necessary. The content includes three sections, i.e., the trigonometric function, concept of distance, scalar, and vector.

Exercises

• 1.1 a = (3, 2, –1), b = (1, –1, 2), please give the following results:

  • (1) $\varvec{a} \times \varvec{b}$

  • (2) $\left( {\varvec{a} + 2\varvec{b}} \right) \times \varvec{b}$

  • (3) $\varvec{a} \cdot 3\varvec{b}$

  • (4) $\varvec{b} \times \varvec{a}$

  • (5) $\varvec{b} \times \left( {3\varvec{a}{-}2\varvec{b}} \right)$

• 1.2 a = (2, –1, 3), b = (1, –1, 3), c = (1, –2, 0), please try to present the final results:

  • (1) $\varvec{a} \times \varvec{b} \cdot \varvec{c}$

  • (2) $\varvec{a} \cdot \varvec{bc}$

  • (3) $\varvec{a} \times \left( {\varvec{b} \times \varvec{c}} \right)$

  • (4) $\left( {\varvec{a} \times \varvec{b}} \right) \times \varvec{c}$

  • (5) $\varvec{c} \times \left( {\varvec{b} \cdot \varvec{c}} \right)\varvec{a}$

• 1.3 A = (1, –2, 3), B = (3, –2, 4), C = (0, –1, 0), and D = (2, –1, 3). Please give the following results:

  • (1) $\varvec{A} \times \varvec{B} \times \varvec{C} - 2\varvec{D}$

  • (2) $\varvec{B} \times \left( {\varvec{A} \times \varvec{C}} \right) - \varvec{D} \times \varvec{A}$

Answers

• 1.1

  • (1) (3, –7, –5)

  • (2) (3, –7, –5)

  • (3) –3

  • (4) (–3, 7, 5)

  • (5) (–9, 21, 15)

• 1.2

  • (1) 6

  • (2) (6, –3, 9)

  • (3) (–8, 20, 12)

  • (4) (–2, –1, 3)

  • (5) (–18, –9, 9)

• 1.3

  • (1) (0, 2, –4)

  • (2) (–1, 18, 9)