Useful Vector Formulae

Vector addition:

$${\bf a}+ {\bf b} \equiv (a_x+b_x,\,a_y+b_y,\, a_z+b_z)$$

Scalar multiplication:

$$n\,{\bf a} \equiv (n\,a_x,n\,a_y, n\,a_z)$$

Scalar product:

$${\bf a}\cdot{\bf b} = a_x \,b_x + a_y \,b_y + a_z\, b_z$$

Vector product:

$${\bf a}\times{\bf b} = (a_y \,b_z-a_z\, b_y,\, a_z\, b_x-a_x\, b_z,\, a_x\, b_y-a_y\, b_x)$$

Scalar triple product:

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

Vector triple product:

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

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