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Vector Products

The concept of vectors as abstract objects representing a collection of data has already been presented. Every student at this point has already encountered vectors as representation of points, forces, and accelerations in two and three dimensions.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

(notebook Lecture-11)

(html Lecture-11)

(xml+mathml Lecture-11)

Solution to 2D Diffusion Equation for Point Source Initial Conditions

$$\vec{J} = -D ( \frac{\partial C}{\partial x} , \frac{\partial C}{\partial y})$$

Review: The Inner (dot) product of two vectors and relation to projection

An inner (or dot-) project is multiplication of two vectors that produces a scalar.

$$\begin{split}\vec{a} \cdot \vec{b} \equiv &\ \equiv & a_i b_... ...ay}{l} a_1\ a_2\ \vdots\ a_N \end{array} \right) \end{split}$$ (11-1)

The inner product is:

linear, distributive

$(k_1 \vec{a} + k_2 \vec{b}) \cdot \vec{c} =$ $k_1 \vec{a} \cdot \vec{c} + k_2 \vec{b} \cdot \vec{c}$

symmetric

$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$

satifies Schwarz inequality

$\norm {\vec{a} \cdot \vec{b}} \leq \norm {\vec{b}} \norm {\vec{a}}$

satifies triangle inequality

$\norm {\vec{a} + \vec{b}} \leq \norm {\vec{b}} + \norm {\vec{a}}$

If the vector components are in a cartesian (i.e., cubic lattice) space, then there is a useful equation for the angle between two vectors:

$$\cos \alpha = \frac{\vec{a} \cdot \vec{b}}{\norm {\vec{a}}\norm {\vec{b}}} = \hat{n_a} \cdot \hat{n_b}$$ (11-2)

where $\hat{n_i}$ is the unit vector that shares a direction with $i$ . Caution: when working with vectors in non-cubic crystal lattices (e.g, tetragonal, hexagonal, etc.) the angle relationship above does not hold. One must convert to a cubic system first to calculate the angles.

The projection of a vector onto a direction $\hat{n_b}$ is a scalar:

$$p = \vec{a} \cdot \hat{n_b}$$ (11-3)

Review: Vector (or cross-) products

The vector product (or cross $\times$ ) differs from the dot (or inner) product in that multiplication produces a vector from two vectors. One might have quite a few choices about how to define such a product, but the following idea proves to be useful (and standard).

normal

Which way should the product vector point? Because two vectors (usually) define a plane, the product vector might as well point away from it.

The exception is when the two vectors are linearly dependent; in this case the product vector will have zero magnitude.

The product vector is normal to the plane defined by the two vectors that make up the product. A plane has two normals, which normal should be picked? By convention, the ``right-hand-rule'' defines which of the two normal should be picked.

magnitude

Given that the product vector points away from the two vectors that make up the product, what should be its magnitude? We already have a rule that gives us the cosine of the angle between two vectors, a rule that gives the sine of the angle between the two vectors would be useful. Therefore, the cross product is defined so that its magnitude for two unit vectors is the sine of the angle between them.

This has the extra utility that the cross product is zero when two vectors are linearly-dependent (i.e., they do not define a plane).

This also has the utility, discussed below, that the triple product will be a scalar quantity equal to the volume of the parallelepiped defined by three vectors.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

(notebook Lecture-11)

(html Lecture-11)

(xml+mathml Lecture-11)

Example of a Cross Product

$$\vec{a} \times \vec{b} = \det \left( \begin{array}{ccc} \hat... ...{k}\\ a_ 1 & a_2 & a_3\\ b_ 1 & b_2 & b_3 \end{array}\right)$$

The triple product,

$$\begin{split}\vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a... ...norm {\vec{c}} \sin \gamma_{a-b} \cos \gamma_{ab-c} \end{split}$$ (11-4)

where $\gamma_{i-j}$ is the angle between two vectors $i$ and $j$ and $\gamma_{ij-k}$ is the angle between the vector $k$ and plane spanned by $i$ and $j$ . is equal to the parallelepiped that has $\vec{a}$ , $\vec{b}$ , and $\vec{c}$ , emanating from its bottom-back corner.

If the triple product is zero, the volume between three vectors is zero and therefore they must be linearly dependent.