Systems of particles -- and that's everything -- are analyzed with vectors. A vector in two dimensions has two numbers associated with it; a vector in three dimensions has three numbers. both the $x$ axis ( abscissa , for more detail) and .
The three black lines are perpindicular, aka Cartesian coordinates. which each act as a ruler for the vector. The two black lines on the page have shadow, and the axis coming out of the page is shadowless. There is one dimension on a measuring tape, .
The axes used here, in three dimensional space are orthogonal which means the $x$-axis is perpindicular to the $y$-axis and also perpindicular to the $z$-axis. This is a convention so you don't see the little square symbol depicted, which means the two lines form a right angle.
A vector is directional and also has a magnitude (or length). We can say the vector is completely described by a set of three numbers, here as $[x=2, y=3, z=2]$.
Vector algebra is when you add two vectors, because subtraction is just addition of the negative. Vector addition happens when you take vector-1 ($v_1$) and add There are other vector operations we will use such as the dot product and the cross product, but you only have the latter in 3-D, so for two dimensions there is only the dot product.