This page assumes nothing. Before you can read the parent note Covariant and contravariant components, you must be fluent in every symbol below. We build them one at a time, each on top of the last, each anchored to a picture.
The picture. Look at the black arrow below. It sits there on a blank sheet. There are no numbers yet — numbers only appear when we choose reference arrows to measure against.
Why the topic needs it. The entire drama of covariant/contravariant is that the arrow is fixed but its numbers change with the grid. If you don't first believe the arrow lives independently of coordinates, none of the two-readings business will make sense.
The picture. Below, e1 and e2 are drawn in lavender and coral. Notice they are not at 90° and not the same length. This is a skew basis — deliberately awkward, so the two-readings effect shows up.
Link forward: choosing a different basis is exactly what Change of basis and transformation laws studies.
Why the topic needs it. Every component statement is built from these two moves. When we write v=v1e1+v2e2, we are saying: scalee1 by the number v1, scalee2 by v2, then add the two scaled arrows to land exactly on the tip of v.
This is the notation that scares people. It is nothing more than where the little label sits.
The picture. Think of a staircase: the lower index sits on the bottom step, the upper index on the top step. The parent note's mnemonic — "CO goes down to the cellar, contra goes up" — is exactly this staircase.
Full index machinery lives in Tensors and index notation.
Why an upper AND a lower? The pairing "one up, one down" is a signal that the result is a genuine coordinate-free number (we'll see why in §8). If you ever see the same letter twice both up or both down, something is wrong.
Everything geometric in this topic — lengths, angles, projections, the metric — is built from one operation: the dot product.
Why this exact formula, and why cos? We want a gadget that (a) grows with length, and (b) tells us how aligned two arrows are. The cosine is the natural "alignment dial":
Arrows pointing the same way → θ=0°, cos0°=1 → biggest value.
Arrows at right angles → θ=90°, cos90°=0 → dot product is zero. (This is how we detect perpendicularity!)
Arrows pointing opposite → θ=180°, cos180°=−1 → most negative.
The picture. In the figure, the coral arrow's shadow falling straight down onto the lavender arrow has length ∣b∣cosθ. The dot product is that shadow length times ∣a∣. This "shadow onto an axis" idea is literally the definition of a covariant component in the parent note.
Now that we have arrows, a basis, scaling, and the dot product, the two readings appear naturally.
The picture. Below, the same v is read both ways. The dashed parallel lines give v1,v2 (walk along the slanted grid). The dotted perpendicular drops give v1,v2 (shadows onto each axis). On this skew grid they differ — on square paper they'd land on top of each other.
Why the topic needs both. Neither reading is "the right one" — they are two faces of the same arrow. The parent note's job is to carry both and translate between them using a small table of dot products called the metric.
Why the topic needs it. The metric is the dictionary between the two readings: vi=gijvj (down the stairs) and vi=gijvj (up the stairs, using the inverse table gij). When the basis is orthonormal, every dot product is a Kronecker delta, so gij=δij, the metric becomes the do-nothing table, and the two readings become identical — which is why square-paper school maths never mentioned any of this.
Deeper study: Metric tensor, and its role in General relativity — raising and lowering indices. The perpendicular-reading also has its own set of reference arrows, the Dual (reciprocal) basis.
