This account of dot product and cross product of vectors advises strongly against introducing both at the same time; but the demands of the A level syllabus mean I’ve had to do that.
They also advise starting with the geometric definition and deriving the algebraic definitions, rather than vice versa. Generally I’m sympathetic to more geometry, less algebra. And it is easier to derive the algebraic definitions from the geometric, rather than vice versa.
But I’ve concluded that for comprehensible sequence in the A level syllabus you just have to start with algebraic.
That said, I think this is a good and clear exposition.