PART 01
对应分量Matching components
ax 只与 bx 相乘，ay 只与 by 相乘。ax pairs with bx; ay pairs with by.
01 · 两个人一起推箱子01 · Two people push a crate
同样用力时，同向、垂直和反向，真正帮上忙的程度会一样吗？With equal effort, do pushing together, sideways, and against each other help by the same amount?
02方向对齐盘Alignment Dial
双箭头旋转型Rotate two arrows
最小夹角Smallest angle
35°
θ
投影长度Projection
2.46
|b| cos θ
点积Dot product
9.83
—
03一次点积的 4 个零件Four Parts of One Dot Product
乘、乘、再相加Multiply, multiply, addPART 02
乘积Products
每一维贡献一个正、零或负的对齐量。Each dimension contributes positive, zero, or negative alignment.
PART 03
求和Sum
把所有维度贡献合成一个标量分数。All dimensional contributions collapse into one scalar score.
PART 04
投影直觉Projection intuition
b 在 a 方向上的影子越长且同向，点积越大。The longer b's same-direction shadow on a, the larger the dot product.
04最小机制：分量算法与夹角直觉是同一件事Minimal Mechanism: Component Math and Angle Intuition Agree
a · b
a · b = axbx + ayby =
|a||b| cos θ
上方固定 |a| = 4、|b| = 3。旋转时只有 cos θ
改变，因此点积随夹角从正到零再到负。Above, |a| = 4 and |b| = 3. Rotation changes only cos θ, so the
dot product moves from positive to zero to negative.
旋转箭头Rotate an arrow夹角 θ 改变Angle θ changes
投影改变Projection changes同向影子缩短或反向The aligned shadow shrinks or reverses
点积改变Dot product changes一个数记录对齐One number records alignment
05边界实验：长度都为正，点积就为正？Boundary Test: Positive Lengths Mean a Positive Dot Product?
主动制造失败Create a failure把 b 旋到完全反向Turn b directly backward
两支箭头长度仍为 4 和 3，都是正数。点击后夹角变 180°，投影反向，点积应为负。Both lengths remain positive at 4 and 3. At 180°, the projection reverses and the dot product must be negative.
尚未执行。长度描述大小，点积还要看方向。Not run yet. Length measures size; a dot product also depends on
direction.
✗ 两个非零向量的点积一定非零。Two nonzero vectors must have a nonzero dot
product.
→ ✓ 垂直向量的点积等于零。Perpendicular vectors have a zero dot product.
✗ 点积越大，只说明两个向量越像。A larger dot product only means greater
similarity.
→ ✓ 点积同时受方向与长度影响。Dot product depends on both direction and length.
✗ 负点积表示计算出错。A negative dot product means the calculation
failed.
→ ✓ 负值表示总体方向超过
90°，更偏反向。A negative value means the directions differ by more than
90°.
06 · 一句话带走06 · One line to keep