Scalar and Vector Products

The scalar product of the vectors a and b is

$${\bf a}\cdot {\bf b} = a_{x}b_{x} + a_{y}b_{y} + a_{z}b_{z} = a_{r}b_{r} + a_{\theta}b_{\theta} + a_{z}b_{z},$$ (1.5)

and, equivalently

$${\bf a}\cdot {\bf b} = a b \cos \theta.$$

Thus the scalar product is independent of the coordinate system.

The vector product is

$${\bf a} \times {\bf b} = ab \sin \theta {\hat{\bf c}},$$ (1.6)

where $\hat{\bf {c}}$ is a unit vector perpendicular to both ${\bf a}$ and ${\bf b}$. The result is invariant with respect to the coordinate system. In cartesian coordinates it is given by

$${\bf a}\times {\bf b} = (a_{y}b_{z} - a_{z}b_{y}){\bf i} + ... ...b_{x} - a_{x}b_{z}){\bf j} + (a_{x}b_{y} - a_{y}b_{x}){\bf k}.$$ (1.7)

The easy way to learn this is to memorise the ${\bf i}$ component and the others are obtained by cyclic rotation of the subscripts,

$$x\hbox{ }\rightarrow\hbox{ }y\hbox{ }\rightarrow\hbox{ }z\hbox{ }\rightarrow\hbox{ }x.$$

The alternative method is to expand the determinant

$${\bf a} \times {\bf b} = \left\vert \begin{array}{ccc} {\... ...y} & a_{z} \\ b_{x} & b_{y} & b_{z} \end{array} \right\vert$$ (1.8)

The triple scalar product is defined as

$${\bf a}\cdot ({\bf b}\times {\bf c}) = {\bf b}\cdot ({\bf c}\times {\bf a}) = {\bf c}\cdot ({\bf a}\times {\bf b})$$ (1.9)

$${\bf a}\cdot ({\bf b}\times {\bf c}) = -{\bf a}\cdot ({\bf ... ...\bf a}\times {\bf c}) = -{\bf c}\cdot ({\bf b}\times {\bf a}).$$

The triple scalar product gives the volume of a parallelopiped formed by sides defined by the vectors ${\bf a},{\bf b}$ and ${\bf c}$.

The triple vector product must be learnt. It is

$${\bf a}\times ({\bf b} \times {\bf c}) = ({\bf a}\cdot {\bf c}){\bf b} - ({\bf a}\cdot {\bf b}){\bf c}.$$ (1.10)

Similarly,

$$({\bf a}\times {\bf b}) \times {\bf c}) = ({\bf a}\cdot {\bf c}){\bf b} - ({\bf b}\cdot {\bf c}){\bf a}.$$