what is a vector
Say "vector" and two pictures come to mind: an arrow, and a pair of numbers like (3, 2). If you have ever wondered which one is the real vector, the answer is both, because they are the same thing. A vector is a single movement: go this far in this direction. Draw that movement and you get an arrow; record how far it goes along each axis and you get a number pair. Hold on to this view and vector addition, along with the matrices that come later, all read the same way. This lesson also pins down î and ĵ, the two arrows that the later lesson "Matrices Are Transformations" uses as its key for reading a matrix.
Read as a movement, (3, 2) is an instruction: go 3 units right and 2 units up. The blue arrow in the figure draws that instruction, and the number pair underneath writes the same instruction in numbers. Drag the arrow's tip and the pair updates in half-unit steps, because the picture and the numbers always describe one and the same movement. In the picture, direction and length are visible at a glance; in numbers, the arithmetic is easy. Use whichever suits the problem, and get comfortable switching between the two. That fluency is the first step.
Adding vectors means chaining movements. Go along a, then from that point go along b, and where you end up is a + b. Each arrow starts where the previous one ends, which is why this is called the tip-to-tail rule. In the figure, the dashed orange segment starts at the tip of the green arrow a, and the blue arrow from the origin to the end of b is the sum. Drag the green dot or the blue dot and the components in the equation below change with it. In numbers it is even simpler: add component by component, so (3,2)+(1,4) is (4,6). Chaining movements in order is all there is to vector addition.
Multiplying a vector by a number keeps it on the same line and changes only the distance. Multiply by 2 and it goes twice as far; multiply by 0.5 and it goes half as far in the same direction. Multiply by a negative number and the length scales by its magnitude, but the direction flips to the exact opposite. Move the slider for s: the gray arrow stays put while the scaled arrow stretches and shrinks along the dashed line. The moment s turns negative, that arrow crosses to the other side and turns red. Multiplying a vector by a number leaves the direction alone and adjusts only the distance.
Now for the central idea of this lesson. Take two basic arrows: î, one unit to the right, and ĵ, one unit up. By stretching and adding just these two, you can reach any point in the plane. (3, 2) takes 3 of î and 2 of ĵ, so it is written 3î + 2ĵ. Drag the blue point and the figure traces the route to it: first along î on the dashed green line from the origin, then up along ĵ on the dashed orange line. The equation below records how many î's and ĵ's the point is built from. Every vector in the plane comes down to one recipe: so much î, so much ĵ. That is exactly why the later lesson "Matrices Are Transformations" can say a matrix is settled once you know where î and ĵ land. Every point is a combination of î and ĵ.
So how far can you get with î and ĵ? The whole plane, and that is what "2D space" means. A space is the set of every point you can reach by stretching and adding two different directions. If the two directions lie on the same line, things change: no amount of stretching and adding gets you off that line. At first the figure shows green u and orange w pointing different ways, and the whole stage is shaded. Press "Make parallel" and w swings onto the line of u; the shaded region collapses to a single band along that line. A space is every point your directions can reach, and only independent directions open up the full plane. This observation, that parallel directions flatten everything, leads straight to the determinant of 0 in the lesson "The Determinant Is Area Scaling."