Vector Magnitude · Same Direction, Different Size

01 · 先看生活直觉 01 · Start with intuition

同一方向走 1 公里和 5 公里，路线朝向没变，什么变了？ Walking 1 km or 5 km in the same direction changes what, if the heading stays fixed?

02 固定方向，拖动长度 Keep Direction Fixed, Drag Magnitude

唯一变量：长度 One variable: magnitude

2.62

横向分量 horizontal component

1.84

纵向分量 vertical component

‖v⃗‖

3.20

原点距离 distance from origin

03 把箭头拆成 4 个零件 Break the Arrow into Four Parts

大小 ≠ 方向size ≠ direction

ORIGIN

原点 Origin

长度从共同起点 (0, 0) 开始量。 Magnitude is measured from the shared start at (0, 0).

COMPONENTS

两个分量 Two components

x 与 y 是同一支箭头在两根轴上的投影。 x and y are projections of the same arrow onto two axes.

MAGNITUDE

整体长度 Overall magnitude

它是直角三角形斜边，不是 x+y。 It is the hypotenuse, not x+y.

UNIT VECTOR

单位向量 Unit vector

归一化只保留方向，把长度缩放到 1。 Normalization preserves direction while scaling magnitude to 1.

04 最小机制：分量平方和再开方 Minimal Mechanism: Square, Add, Take the Root

Pythagoras

‖v⃗‖ = √(x² + y²)

归一化：v̂ = v⃗ / ‖v⃗‖。上方长度非零时，点击按钮即可验证结果恒为 1。 Normalization: v̂ = v⃗ / ‖v⃗‖. When magnitude is nonzero, use the button above to verify the result becomes 1.

拖动长度 Drag magnitude 端点沿固定射线移动 endpoint moves along one ray

分量同比缩放 Components scale together x = r cos θ · y = r sin θ

距离变为 rDistance becomes r 方向 θ 仍是 35° direction θ remains 35°

05 边界实验：零向量不能归一化 Boundary Test: The Zero Vector Cannot Be Normalized

主动制造失败 Create a failure

把长度压到 0，再除以它 Set magnitude to 0, then divide by it

零向量没有可保留的方向，公式还会出现除以 0。 The zero vector has no direction to preserve, and the formula would divide by zero.

尚未执行。先观察普通向量的归一化。 Not run yet. Normalize a nonzero vector first.

✗ 向量长度就是各分量相加。 Magnitude is the sum of components. → ✓ 长度是平方和的平方根。 Magnitude is the square root of the sum of squares.

✗ 归一化会改变方向。 Normalization changes direction. → ✓ 非零向量只改变尺度，方向保留。 For nonzero vectors it changes scale and preserves direction.

✗ 任何向量都能除以自身长度。 Every vector can be divided by its magnitude. → ✓ 零向量必须单独处理。 The zero vector requires a separate case.

06 · 一句话带走 06 · One line to keep

向量长度是从原点到端点的距离，表达整体大小而非方向；拖回上方可验证同一方向能有不同长度。 Vector magnitude is the distance from origin to endpoint. It expresses overall size, not direction; drag above to verify that one direction can have many magnitudes.