Vector · An Ordered Set of Coordinates

01 · 空间里的位置01 · A position in space

只说“向右 3”够不够，还是还要交代向上和向前各多少？ Is “three units right” enough, or must you also say how far up and forward?

023D 向量坐标台3D Vector Coordinate Bench

端点拖拽型Endpoint drag

拖红色端点，改变整支向量Drag the red endpoint to change the vector

红色箭头端点是唯一主变量。拖动它会同时更新三个有序分量；拖空白处只旋转相机。虚线把端点投影回 x、y、z 三轴。 The red endpoint is the single main variable. Drag it to update three ordered components; drag empty space to orbit. Dashed guides project the endpoint onto x, y, and z.

有序分量 [x, y, z]Ordered components [x, y, z] [3.0, 2.0, 2.5]

红色 · x 分量Red · x component3.0

蓝色 · y 分量Blue · y component2.0

琥珀 · z 分量Amber · z component2.5

固定种子Fixed seed20260722

教学模拟值，非真实模型输出simulated for teaching — not real model output

分量数Components

固定顺序fixed order

当前向量Current vector

3, 2, 2.5

x · y · z

对象位置Object position

箭头端点arrow endpoint

x · 特征 1x · feature 1

y · 特征 2y · feature 2

z · 特征 3z · feature 3

x = 3.0

y = 2.0

z = 2.5

A [3.0, 2.0, 2.5]

1 单位参照1-unit reference

拖红点改向量 · 拖空白旋转 · 滚轮缩放Drag red point to edit · drag space to orbit · wheel to zoom

正在加载 3D；若此提示持续显示，需要 WebGL 或 three.js 未加载。 Loading 3D. If this remains visible, WebGL is required or three.js did not load.

03一支 3D 向量的 4 个零件Four Parts of a 3D Vector

顺序决定含义Order carries meaning

PART 01

三个分量Three components

x、y、z 各是一个标量，合起来描述三维位置。x, y, and z are scalars that together describe a 3D position.

PART 02

固定顺序Fixed order

第一格始终是 x，第二格是 y，第三格是 z；交换就改变含义。The first slot is x, the second y, and the third z; swapping changes meaning.

PART 03

轴上投影Axis projections

虚线把同一端点拆回三轴位置，读出每个分量。Dashed guides split one endpoint back onto three axes so each component can be read.

PART 04

方向与位置Direction and position

从原点到端点的整支箭头，同时给出整体方向和对象位置。The full arrow from origin to endpoint gives both overall direction and object position.

04最小机制：按顺序把标量装成一组Minimal Mechanism: Pack Scalars in Order

v = [v₁, v₂, v₃]

v = [x, y, z] ∈ R³

拖动端点时，x、y、z 同步更新；直接改任一分量，端点会移动到对应坐标。 Drag the endpoint to update x, y, and z, or edit a component to move the endpoint to that coordinate.

移动端点Move the endpoint 对象位置改变Object position changes

投影更新Projections update 三轴读数改变Three axis readings change

有序数列更新Ordered list updates 表示随位置改变Representation follows position

05边界实验：交换分量却假装没变Boundary Test: Swap Components and Pretend Nothing Changed

主动制造失败Create a failure

把 [x, y, z] 写成 [y, x, z]Write [y, x, z] instead of [x, y, z]

点击后交换前两个数。数值集合没变，但坐标角色改变，箭头端点会跳到另一个位置。Swap the first two numbers. The set of values stays the same, but their roles change and the endpoint jumps.

尚未执行。向量不是一袋无序数字。Not run yet. A vector is not an unordered bag of numbers.

✗ [3, 2, 2.5] 与 [2, 3, 2.5] 表示同一位置。[3, 2, 2.5] and [2, 3, 2.5] represent the same position. → ✓ 分量顺序绑定维度含义。Component order binds each dimension's meaning.

✗ 向量只是画出来的一支箭头。A vector is merely a drawn arrow. → ✓ 箭头是有序数列的几何表示。The arrow is a geometric view of an ordered list.

✗ 旋转相机就改变了向量。Orbiting the camera changes the vector. → ✓ 观察角度改变，底层三个数不变。The view changes; the three underlying numbers do not.

06 · 一句话带走06 · One line to keep

向量是按固定顺序排列的一组数，用多个维度共同定位对象；拖动 3D 端点，三轴投影、坐标、方向和位置会一起变化。 A vector is an ordered list of numbers that locates an object across dimensions. Drag the 3D endpoint and its axis projections, coordinates, direction, and position move together.