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Advanced Engineering Dynamics

R. Valéry Roy

Section A.1 Vector Algebra

Subsection A.1.1 Vectors

We can define vectors in an intuitive manner as a directed line segment. Hence, a vector \(\bU\)  1  is characterized by its magnitude, denoted \(|\bU|\text{,}\) and by its direction. On a more abstract level, vectors are defined as equivalent classes of ordered pairs of points \((A,B)\) of a three-dimensional space: two ordered pairs \((A,B)\) and \((C,D)\) are equipollent if (i) their supports are parallel, (ii) they have the same order, (iii) they have the same magnitude. The equivalence \((A,B)\sim (C,D)\) is denoted \(\overrightarrow{AB}= \overrightarrow{CD}\text{.}\) All ordered pairs equivalent to \((A,B)\) are then defined as vector \(\bU = \overrightarrow{AB}\text{.}\) See Figure A.1.1.
Hence two vectors \(\bU\) and \(\bV\) are equal if and only if they are parallel, have the same direction and the same magnitude: we then write \(\bU =\bV\text{.}\) The starting points of vectors are thus immaterial.
Figure A.1.1.
We can then define two operations which make the set of vectors a vector space \(E\) over the set of real numbers \(\mathbb{R}\text{:}\)
  1. Multiplication by a scalar (the identity element is the real number \(1\text{.:}\) the product of scalar \(\lambda\) and vector \(\bU\) is a vector parallel to \(\bU\text{,}\) of magnitude \(|\la| |\bU|\text{,}\) in the same direction as \(\bU\) if \(\la \gt 0\text{,}\) opposed to \(\bU\) if \(\la \lt 0\text{.}\)
  2. Addition of vectors (the identity element is the zero vector \(\bze\text{.:}\) the sum of two vectors \(\bU\) and \(\bV\) is obtained by constructing a triangle with \(\bU\) and \(\bV\) forming two sides, \(\bV\) adjoined to \(\bU\text{:}\) then the sum \(\bU+\bV\) is the vector starting at the origin of \(\bU\) and ending at the arrow of \(\bV\text{.}\) See Figure A.1.2.
To qualify as a vector space, the set \(E\) and the operations of addition and multiplication must adhere to the following axioms: given three vectors \(\bU\text{,}\) \(\bV\text{,}\) and \(\bW\text{,}\) and two real scalars \(\la\) and \(\mu\text{,}\)
\begin{align*} \bU+\bV = \bV+\bU \amp \qquad \amp 1 \bU = \bU\\ (\bU+\bV)+ \bW = \bU+(\bV+\bW) \amp \qquad \amp \la (\bU+\bV) = \la \bU + \la \bV\\ \bU + \bze = \bU \amp \qquad \amp (\la+\mu)\bU = \la \bU + \mu \bU\\ \bU + (-\bU) = \bze \amp \qquad \amp \la(\mu \bU) = \la\mu \bU \end{align*}
Figure A.1.2.

Definition A.1.3. Collinear vectors.

Two vectors \(\bU\) and \(\bV\) are said to be collinear if there exists a non-zero scalar \(\la\) such that \(\bV =\la \bU\)

Definition A.1.4. Linear Independent vectors.

Two vectors \(\bU\) and \(\bV\) are said to be linearly independent if \(\la \bU +\mu \bV =\bze\) implies \(\la=\mu = 0\text{.}\)

Definition A.1.5. Basis of vectors.

Three vectors \((\bU_1, \bU_2,\bU_3)\) are said to form a basis of \(E\) if they are linearly independent. Then any vector \(\bV \in E\) can be written uniquely in the form
\begin{equation*} \bV= V_1 \bU_1 + V_2 \bU_2 + V_3 \bU_3 \end{equation*}
The scalars \(V_1\text{,}\) \(V_2\) and \(V_3\) are the components of \(\bV\) on basis \((\bU_1, \bU_2,\bU_3)\text{.}\)

Subsection A.1.2 Scalar Product

Definition A.1.6. Scalar product.

The scalar product (or dot product) between vectors \(\bU\) and \(\bV\) is the scalar defined by \(\bU \cdot \bV = | \bU | | \bV | \cos \al\text{,}\) where \(\al \in [0 , \pi] \) is the angle measured between \(\bU\) and \(\bV\text{.}\)

Properties A.1.1.

  1. Distributivity: \(\bU \cdot (\bV + \bW ) = \bU \cdot\bV + \bV \cdot \bW\)
  2. Multiplication by a scalar \(\la\text{:}\) \(\bU \cdot ( \la \bV) = \la \bU \cdot \bV\)
  3. If \(\bU \cdot \bV = 0\text{,}\) the non-zero vectors \(\bU\) and \(\bV\) are said to be orthogonal.
  4. The scalar \(|\bU | = (\bU \cdot \bU)^{1/2}\) is the magnitude or normnorm of \(\bU\text{.}\)  2 .
  5. The orthogonal projection of vector \(\bU\) on the line \(\Delta =\{\la \bV |\la \in \mathbb{R}\}\) is the vector (see Figure A.1.7)
    \begin{equation*} \text{proj}_\bV (\bU) = \frac{\bU \cdot \bV}{\bV^2} \bV \end{equation*}
  6. The orthogonal projection of vector \(\bW\) on the plane spanned by two linear independent vectors \(\bU\) and \(\bV\) is the vector
    \begin{equation*} \text{proj}_{(\bU,\bV)} (\bW) = \frac{\bW \cdot \bU}{\bU^2} \bU + \frac{\bW \cdot \bV}{\bV^2} \bV \end{equation*}
  7. A unit vector is a vector of magnitude \(1\) and denoted \(\bu\text{.}\)
  8. A orthonormal basis \(b (\be_1, \be_2, \be_3)\) is a basis satisfying \(\be_i \cdot \be_j = 0\) if \(i\neq j\) and \(\be_i \cdot \be_i = 1\text{.}\) Vector space \(E\) equipped with the scalar product is a Euclidean space of dimension \(3\text{.}\) Any vector \(\bU\) of \(E\) can then be written on basis \(b\) as
    \begin{equation*} \bU = (\be_1 \cdot \bU) \be_1 + (\be_2 \cdot \bU) \be_2 + (\be_3 \cdot \bU) \be_3 \end{equation*}
  9. Consider another basis \(b' (\be_1',\be_2',\be'_3)\) of \(E\) and define the coefficients \(c_{ij}= \be_i \cdot \be'_j\text{.}\) Consider the \(3\times 3\) matrix \([\cC]= [c_{ij}]\text{:}\) the \(j\)th column of \([\cC]\) is composed of the components of \(\be_j'\) on basis \(b\text{.}\) It is easily seen that \([\cC]\) gives the components of vector \(\bU\) from one basis to the other: more specifically, if \(\bU = \sum U_i \be_i = \sum U'_i \be_i' \) then
    \begin{equation*} U_i = \sum c_{ij} U'_j \end{equation*}

Subsection A.1.3 Cross Product

Definition A.1.8. Cross product of vectors.

The cross product between two vectors \(\bU\) and \(\bV\) is the vector denoted \(\bU \times \bV\) such that (see Figure A.1.9):
  • \(\bU \times \bV\) is orthogonal to the plane spanned by \(\bU\) and \(\bV\text{,}\)
  • \((\bU , \bV , \bU \times \bV)\) is oriented in the right-handed direction  3 ,
  • its magnitude is given by \(| \bU \times \bV | = | \bU | | \bV | \sin \al \text{,}\) where \(\al \in [0, \pi] \) is the angle between \(\bU\) and \(\bV\text{.}\)
Figure A.1.9.

Properties A.1.2.

  1. Skew-symmetry: \(\bU \times \bV = - \bV \times \bU\text{,}\) and hence \(\bU \times \bU = {\bf 0}\)
  2. Distributivity: \(\bU \times (\bV + \bW ) = \bU \times \bV + \bV \times \bW\)
  3. Multiplication by a scalar \(\la\text{:}\) \(\bU \times ( \la \bV) = \la \bU \times \bV\)
  4. If three vectors satisfy the equality \(\bU \times \bV = \bU \times \bW \text{,}\) then \(\bV = \bW + \la \bU\) where \(\la\) is an indeterminate scalar.
  5. Given a right-handed orthonormal basis \((\be_1 , \be_2 , \be_3)\text{,}\) we have the following relationships
    \begin{equation*} \be_1 = \be_2 \times \be_3, \qquad \be_2 = \be_3 \times \be_1, \qquad \be_3 = \be_1 \times \be_2 \end{equation*}
    Then, the cross product between \(\bU = U_1 \be_1 +U_2 \be_2 +U_3 \be_3\) and \(\bV = V_1 \be_1 +V_2 \bV_2 +V_3 \be_3\) can be calculated according to
    \begin{equation*} \bU \times \bV = \det\begin{pmatrix} \be_1 \amp \be_2 \amp \be_3 \\ U_1 \amp U_2 \amp U_3 \\ V_1 \amp V_2 \amp V_3 \end{pmatrix} = (U_2 V_3 - U_3 V_2) \be_1 + (U_3 V_1 - U_1 V_3) \be_2 + (U_1 V_2 - U_2 V_1) \be_3 \end{equation*}
  6. \(| \bU \times \bV |\) represents the area of the parallelogram formed by the two vectors.
  7. Given three vectors \(\bU\text{,}\) \(\bV\) and \(\bW\text{,}\) the triple vector product \(\bU \times (\bV \times \bW)\) can be found according to the formula
    \begin{equation*} \bU \times (\bV \times \bW) = (\bU \cdot \bW) \bV - (\bU \cdot \bV) \bW \end{equation*}
    In general, \(\bU \times (\bV \times \bW)\) is not equal to \((\bU \times \bV ) \times \bW\text{.}\) In fact, we have
    \begin{equation*} \bU \times (\bV \times \bW) = (\bU \times \bV ) \times \bW + \bV \times (\bU \times \bW ) \end{equation*}
  8. Given two vectors \(\bU\) and \(\bV\) we have (Lagrange formula)
    \begin{equation*} (\bU \cdot \bV)^2 + (\bU \times \bV)^2 = \bU^2 \, \bV^2 \end{equation*}

Subsection A.1.4 Triple Scalar Product

Definition A.1.10. Triple Scalar Product.

The triple scalar product between three vectors \(\bU\text{,}\) \(\bV\text{,}\) and \(\bW\) is the scalar denoted \((\bU , \bV , \bW)\) and defined as
\begin{equation*} (\bU , \bV , \bW ) = (\bU \times \bV) \cdot \bW \end{equation*}

Properties A.1.3.

  1. The triple scalar product \((\bU , \bV , \bW )\) is the volume of the parallelepiped formed by the three vectors.
  2. Under circular permutation of the vectors, the triple scalar product remains unchanged
    \begin{equation*} (\bU , \bV , \bW ) = (\bV , \bW , \bU ) = (\bW , \bU , \bV ) \end{equation*}
  3. The triple scalar product changes sign under the permutation of any two vectors,
    \begin{equation*} (\bU , \bV , \bW ) = -(\bV , \bU , \bW) = - (\bU , \bW , \bV ) \end{equation*}
  4. Given three non-zero vectors, \((\bU , \bV , \bW )\) is zero if two of the three vectors are collinear. Conversely, if the triple product \((\bU , \bV , \bW)\) is zero, two of the vectors are collinear or all three vectors are coplanar.
  5. The triple scalar product can be calculated by resolving each vector on a right-handed orthonormal basis \((\be_1 , \be_2 , \be_3)\text{,}\) that is, \(\bU = U_1 \be_1 +U_2 \be_2 +U_3 \be_3\text{,}\) \(\bV = V_1 \be_1 +V_2 \be_2 +V_3 \be_3\text{,}\) and \(\bW = W_1 \be_1 +W_2 \be_2 +W_3 \be_3\text{.}\) Then, we have
    \begin{equation*} (\bU , \bV , \bW ) = \left| \begin{array}{ccc} U_1 \amp V_1 \amp W_1 \\ U_2 \amp V_2 \amp W_2 \\ U_3 \amp V_3 \amp W_3 \end{array} \right| \end{equation*}
  6. The following identity can be proven
    \begin{equation*} (\bU \times \bV ) \cdot (\bW \times \bX ) = (\bU \cdot \bW) (\bV \cdot \bX) - (\bU \cdot \bX) (\bV \cdot \bW) \end{equation*}

Subsection A.1.5 Linear Operators

Definition A.1.11. Linear operator.

An operator \(\cL: E \to E\) is linearlinear if the following two conditions are satisfied:
\begin{equation*} \cL(\la \bU) = \la \cL(\bU) \end{equation*}
\begin{equation*} \cL(\bU+ \bV) = \cL(\bU)+\cL(\bV) \end{equation*}
for any \(\la \in \mathbb{R}\) and any two vectors \(\bU\) and \(\bV\text{.}\) The identity operator \(\cI\) is the operator satisfying \(\cI(\bU) = \bU\text{.}\)

Definition A.1.12. Invertible linear operator.

A linear operator \(\cL\) is said to be invertible if there exists an operator denoted \(\cL^{-1}\text{,}\) called the inverse of \(\cL\text{,}\) such that \(\cL^{-1} \circ \cL = \cL\circ \cL^{-1} = \cI\text{.}\)

Properties A.1.4.

  1. If \(\cL_1\) and \(\cL_2\) of \(E\) are two linear operators, then \(\cL_1 \circ \cL_2\) is a linear operator. In general \(\cL_1 \circ \cL_2 \neq\cL_2 \circ \cL_1\text{.}\)
  2. All linear operators satisfy \(\cL(\bze) = \bze\text{.}\)
  3. The kernel of linear operator \(\cL\) is the subset \(\text{ker}(\cL)= \{ \bX\in E \,|\, \cL(\bX) = \bze\}\text{.}\) The range of \(\cL\) is the subset \(\cL(E)\text{.}\) In general, we have \(\text{dim}(\text{ker}(\cL)) +\text{dim}(\cL(E)) = \text{dim} (E)\text{.}\)
  4. A linear operator \(\cL\) is entirely defined by the mapping by \(\cL\) of basis vectors. Given a right-handed orthonormal basis \(b (\be_1,\be_2,\be_3)\text{,}\) let \(\la_{ij}\) be the scalars defined by
    \begin{equation*} \cL (\be_j) = \sum_{i=1}^3 \la_{ij} \be_i\qquad (j=1,2,3) \end{equation*}
    The scalars \((\la_{ij})_{i,j=1,2,3}\) define the matrix of \(\cL\) on basis \(b\text{:}\) it is denoted \([\cL]_b\) and written in the form of a \(3\times 3\) array
    \begin{equation*} [\cL]_b = \begin{pmatrix} \la_{11} \amp \la_{12} \amp \la_{13} \\ \la_{21} \amp \la_{22} \amp \la_{23} \\ \la_{31} \amp \la_{32} \amp \la_{33} \end{pmatrix}_b \end{equation*}
    The \(j\)th column of matrix \([\cL]_b\) represents the components of vector \(\cL(\be_j)\) on basis \(b\text{.}\) The mapping by \(\cL\) of vector \(\bU\) can then be obtained as the product \([\cL]_b [\bU]_b\) where \([\bU]_b\) represent vector \(\bU\) on basis \(b\text{:}\)
    \begin{equation*} \cL(\bU) = [\cL]_b [\bU]_b = \begin{pmatrix} \la_{11} \amp \la_{12} \amp \la_{13} \\ \la_{21} \amp \la_{22} \amp \la_{23} \\ \la_{31} \amp \la_{32} \amp \la_{33} \end{pmatrix}_b \begin{pmatrix} U_1\\U_2\\U_3 \end{pmatrix}_b =\sum_{i=1}^3 \sum_{j=1}^3 \la_{ij}U_j \be_i \end{equation*}
  5. Given two linear operators \(\cL_1\) and \(\cL_2\) and a basis \(b\text{,}\) we have
    \begin{equation*} \left[\cL_1 \circ \cL_2\right]_b = \left[\cL_1\right]_b\left[\cL_2\right]_b \end{equation*}

Definition A.1.13. Adjoint operator.

The adjoint \(\cL^*\) of operator \(\cL\) of \(E\) is the linear operator satisfying
\begin{equation*} \bU \cdot \cL (\bV) = \cL^*(\bU) \cdot \bV \end{equation*}
for all vectors \(\bU\) and \(\bV\) of \(E\text{.}\) The operator \(\cL\) is said to be symmetric} if \(\cL^*= \cL\text{.}\)

Definition A.1.14. Symmetric and skew-symmetric parts of alinear operator.

The symmetric and skew-symmetric parts of linear operator \(\cL: E \to E\) are the operators defined respectively by
\begin{equation*} \cL_+ = \tfrac{1}{2} (\cL + \cL^*), \qquad \cL_- = \tfrac{1}{2} (\cL - \cL^*) \end{equation*}

Properties A.1.5.

  1. The matrix representation of \(\cL^*\) on basis \(b\) is the transpose of matrix \([\cL]_b^T\text{:}\)
    \begin{equation*} \la^*_{ij} = \la_{ji} \end{equation*}
  2. It is easy to prove that
    \begin{equation*} \cL = (\cL_++ \cL_-), \qquad \cL_+^* = \cL_+, \qquad \cL_-^* = - \cL_- \end{equation*}

Definition A.1.15. Trace of a linear operator.

The trace of a linear operator \(\cL\text{,}\) denoted \(\text{tr}(\cL)\text{,}\) is the scalar \(\la_{11}+\la_{22}+\la_{33}\text{,}\) where \(\la_{ii} = \be_i \cdot \cL(\be_i)\text{.}\) It is independent of the choice of basis \(b\text{.}\)

Properties A.1.6.

  1. The trace satisfies the property \(\text{tr}(\cL_1 \circ \cL_2) =\text{tr}(\cL_2 \circ \cL_1)\text{.}\)
  2. Given an invertible linear operator \(\cL_1\text{,}\) \(\text{tr}(\cL_1^{-1} \circ \cL_2 \circ \cL_1 ) = \text{tr}(\cL_2)\text{.}\)

Definition A.1.16. Orthogonal linear operator.

A linear operator \(\cL\) of \(E\) is said to be orthogonal if it satisfies one the following three equivalent properties:
  1. \(\cL(\bU) \cdot \cL (\bV) = \bU \cdot \bV\) for all \(\bU\) and \(\bV\) of \(E\text{,}\)
  2. \(|\cL(\bU)| = |\bU|\) for all \(\bU\) of \(E\text{,}\)
  3. \(\cL^* \circ \cL = \cI\text{.}\)
The set of orthogonal operators forms a group, called orthogonal group of \(E\text{.}\)

Properties A.1.7.

  1. Given an orthonormal basis \(b\text{,}\) the matrix of \(\cL\) in basis \(b\) is orthogonal}, that is, it satisfies \([\cL]^T_b [\cL]_b = [\cI]_b\text{.}\)
  2. Orthogonal operators satisfy the property \(\det [\cL]_b = \pm 1\text{.}\) The orthogonal operators satisfying \(\det [\cL]_b = 1\) form a subgroup denoted \(SO(3)\text{.}\)
  3. For each orthogonal operator \(\cL\text{,}\) there exists an orthonormal basis \(b(\be_1,\be_2,\be_3)\) such that the matrix of \(\cL\) on \(b\) takes the form
    \begin{equation*} [\cL]_b = \begin{pmatrix} \cos\te \amp -\sin\te \amp 0 \\ \sin\te \amp \cos\te \amp 0 \\ 0 \amp 0 \amp \ep \end{pmatrix}_b \end{equation*}
    with \(\ep = \pm 1\text{.}\) For \(\ep =1\text{,}\) \(\det[\cL]_b = 1\) and \(\cL\) is a rotation} about \(\be_3\) of angle \(\te\text{,}\) denoted \(\cR_{\te,\be_3}\text{.}\) Note that \(\text{tr}(\cR_{\te,\be_3}) = 2\cos\te +1 \text{.}\)
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