You basically only need this for vectors in ##\mathbb{R}^3##.
##\vec u = (u_x, u_y, u_z)##
Definition of scalar product ## \vec u \cdot \vec v = u_xv_y + u_yv_y + u_zv_z = |\vec u | |\vec v | \cos \theta ##, where ##\theta## is the angle bewteen the two vectors and ##|\vec u | = \sqrt{u_x^2 + u_y^2 + u_z^2}## is the norm (lenght) of a vector in R^3.
Definition of cross product ##\vec u \times \vec v = \vec w ## where ##w_x=u_y v_z - u_zv_y##, ##w_y = u_z v_x - u_xv_z## and ##w_z = u_x v_y - u_yv_x##.
As a first set of excersices, show that
1) ## \vec u \cdot \vec u = |\vec u |^2##
2) ## \vec u \cdot \vec v = |\vec u | |\vec v | \cos \theta ##, the law of cosines is helpful here.
3) ##\vec u \times \vec v = - (\vec v \times \vec u)## (the cross product is anti-commutative)
4) ##\vec u \times \vec u = \vec 0## where ##\vec 0 = (0, 0, 0)## is the zero-vector in ##\mathbb{R}^3##.
5) ## \vec u \times (\vec v + \vec w ) = \vec u \times\vec v + \vec u \times \vec w ##
6) ## (k \vec u) \times \vec v = \vec u \times (k\vec v) = k(\vec u\times \vec v) ## where ##k## is a scalar
7) ##\vec u \times (\vec v \times \vec w) \neq (\vec u \times \vec v ) \times \vec w ## (the cross product is non-associative)
8) ## \vec u \cdot (\vec u \times \vec v) = 0 ## and ##\vec v \cdot (\vec u \times \vec v) = 0 ## (the vector produced by the cross product is orthogonal to both of its input-vectors)
9) ##| \vec u \times \vec v| = |\vec u | |\vec v | \sin \theta##
10) ##\vec u \times (\vec v \times \vec w) + \vec v \times (\vec w \times \vec u) + \vec w \times (\vec u \times \vec v) = 0## (Jacobi identity)
You should be able to prove these with just basic high-school algebra such as factorization and some basic trigonometry. I.e. no fancy math required. When you can do these on your own, everything else should be piece of cake.
