// Shared hand-written sample lesson exercising all 4 kinds.
// Content field values are plain content blocks — exactly what the Rust driver
// would hand us via `include`.

#let info = (
  title: "向量与几何 · 示例讲义",
  author: ("张老师", "李老师"),
)

#let parts = (
  // segment — with NESTED headings so per-level numbering is visible:
  //   level-1 `=` should render `一、`, level-2 `==` should render `1.1`
  //   (NOT the old buggy `二、一、`).
  (
    kind: "segment",
    textbook: [
      = 平面向量的数量积

      本节研究平面向量的基本运算。设 $arrow(a)$、$arrow(b)$ 为平面内两个向量，
      其数量积定义为 $arrow(a) dot arrow(b) = |arrow(a)| |arrow(b)| cos theta$，
      其中 $theta$ 为两向量的夹角。

      == 坐标表示

      在直角坐标系下，若 $arrow(a) = (x_1, y_1)$、$arrow(b) = (x_2, y_2)$，则
      $arrow(a) dot arrow(b) = x_1 x_2 + y_1 y_2$。

      == 几何意义

      数量积等于一个向量的模与另一向量在其方向上投影之积。
    ],
  ),
  // example WITH source
  (
    kind: "example",
    source: "2024 高考甲卷",
    problem: [
      已知 $arrow(a) = (1, 2)$，$arrow(b) = (3, -1)$，求 $arrow(a) dot arrow(b)$。
    ],
    solution: [
      由坐标公式，$arrow(a) dot arrow(b) = 1 times 3 + 2 times (-1) = 3 - 2 = 1$。
    ],
  ),
  // example WITHOUT source
  (
    kind: "example",
    problem: [
      求向量 $arrow(a) = (3, 4)$ 的模长 $|arrow(a)|$。
    ],
    solution: [
      $|arrow(a)| = sqrt(3^2 + 4^2) = sqrt(25) = 5$。
    ],
  ),
  // lemma WITH proof
  (
    kind: "lemma",
    stmt: [
      对任意向量 $arrow(a)$、$arrow(b)$，有 $|arrow(a) dot arrow(b)| <= |arrow(a)| |arrow(b)|$。
    ],
    proof: [
      由数量积定义 $arrow(a) dot arrow(b) = |arrow(a)| |arrow(b)| cos theta$，
      又 $|cos theta| <= 1$，故 $|arrow(a) dot arrow(b)| = |arrow(a)| |arrow(b)| |cos theta| <= |arrow(a)| |arrow(b)|$。
      $qed$
    ],
  ),
  // lemma WITHOUT proof (proof key omitted entirely)
  (
    kind: "lemma",
    stmt: [
      两个非零向量垂直当且仅当其数量积为零，即 $arrow(a) perp arrow(b) <==> arrow(a) dot arrow(b) = 0$。
    ],
  ),
  // sop
  (
    kind: "sop",
    sop: [
      求两向量夹角的标准步骤：
      + 计算数量积 $arrow(a) dot arrow(b)$；
      + 分别计算模长 $|arrow(a)|$、$|arrow(b)|$；
      + 代入 $cos theta = (arrow(a) dot arrow(b)) / (|arrow(a)| |arrow(b)|)$ 求出 $theta$。
    ],
  ),
)