Vector algebra

In the section on linear algebra, different symbols were used for vectors and their representation in a particular bases. In this section, we deal only with the vectors themselves. When vectors need to be expressed in a basis, \(3\times 1\) matrices are used. It is therefore not necessary to distinguish between \(\vec{v}\) and \(\mathsf{v}\) (and between \(A\) and \(\mathsf{A}\)).

Some of the problems below come from chapter 1 of Introduction to Electrodynamics by D. J. Griffiths.

1. Suppose we have a barrel of fruit that contains \(a_x\) bananas, \(a_y\) pears, and \(a_z\) apples. Denoting \(\vec{e}_n\) as the unit vector in the \(n\) direction in space, is \(\vec{a} = a_x\vec{e}_x + a_y\vec{e}_y + a_z\vec{e}_z\), a vector? Explain. [answer]

   No, because \(\vec{a}\) does not obey coordinate transformations. For example, choosing a different set of axes does not turn a pear into a banana. By definition, "a vector is any set of three components that transforms in the same manner as a displacement when you change coordinates." (from Griffiths Introduction to Electrodynamics, section 1.1.5).

2. How do the components \(a_x\), \(a_y\), and \(a_z\) of a vector \(\vec{a} = a_x \vec{e}_x + a_y \vec{e}_y + a_z \vec{e}_z \) transform under the translation of coordinates? \begin{align*} x' &= x\\ y' &= y-a\\ z' &= z \end{align*} In other words, what happens to \(a_x\), \(a_y\), and \(a_z\) when \(\vec{a}\) is written as \(\vec{a} = a_x \vec{e}_x' + a_y \vec{e}_y' + a_z \vec{e}_z' \)? [answer]

   The components of a vector are invariant under this transformation.

3. How do the components of a vector transform under the inversion of coordinates? \begin{align*} x' &= -x\\ y' &= -y\\ z' &= -z \end{align*} In other words, what happens to \(a_x\), \(a_y\), and \(a_z\) when \(\vec{a}\) is written as \(\vec{a} = a_x \vec{e}_x' + a_y \vec{e}_y' + a_z \vec{e}_z' \)? [answer]

   The components are also inverted. \(a_x\mapsto -a_x\), \(a_y\mapsto -a_y\), and \(a_z\mapsto -a_y\).

4. How does the cross product of two vectors \(\vec{u}\) and \(\vec{v}\) transform under the inversion of coordinates? Is the cross product of two vectors really a vector? [answer]

   The cross product \(\vec{w} = \vec{u} \times \vec{v}\) is invariant under the inversion, because \(\vec{w} = -\vec{u} \times -\vec{v}\). Thus \(\vec{w}\) is a different kind of quantity than vectors \(\vec{u}\) and \(\vec{w}\). It is called a psuedovector.

5. How does the scalar triple product of \(\vec{w}\cdot(\vec{u} \times \vec{v})\) transform under the inversion of coordinates? Is the scalar triple product really a scalar? (Griffiths problem 1.10d) [answer]

   The scalar triple product transforms as \(-\vec{w}\cdot(-\vec{u} \times -\vec{v}) = -\vec{w}\cdot(\vec{u} \times \vec{v})\), i.e., the product changes signs when the coordinates are inverted. This is in contrast with the fact that scalars are invariant under coordinate inversions. Thus the scalar triple product is a different kind of quantity than an ordinary scalar. It is called a psuedoscalar.

6. Show that \( |\vec{u}\times \vec{v}|^2 + (\vec{u}\cdot \vec{v})^2 = |\vec{u}|^2|\vec{v}|^2 \). [answer]

   \begin{align*} |\vec{u}\times \vec{v}|^2 + (\vec{u}\cdot \vec{v})^2 &=\epsilon_{ijk}u_j v_k \epsilon_{ilm}u_l v_m + u_i v_i u_j v_j\\ &= \epsilon_{ijk}\epsilon_{ilm}u_j v_k u_l v_m + u_i v_i u_j v_j\\ &=(\delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl})u_j v_k u_l v_m + u_i v_i u_j v_j\\ &=\delta_{jl}\delta_{km}u_j v_k u_l v_m -\delta_{jm}\delta_{kl}u_j v_k u_l v_m + u_i v_i u_j v_j \\ &=u_l u_l v_k v_k - u_m v_m v_k u_k + u_i v_i u_j v_j \\ &=u_l u_l v_k v_k - u_i v_i u_j v_j + u_i v_i u_j v_j \\ &=|\vec{u}|^2|\vec{v}|^2 \end{align*}