T.10 · Vector geometry, projections and oriented area
GRE Subject Test · GRE Mathematics Topic 24 2:58 English narration · English + 中文 subtitles burned in
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This unit uses dot products to classify angles and compute projections, calculates triangle area and orientation with a cross product, and constructs or rules out planar dot-product sign configurations.
本单元用点积判断角度并计算投影，用叉积计算三角形面积和定向，并构造或排除平面向量点积符号的各种配置。
The dot product is the product of lengths times cos theta, so its sign classifies the angle.
点积等于两长度之积乘以 cos θ，所以它的符号决定角的类型。
The vector projection is v dot u over u dot u, times u, and the remainder is orthogonal to u.
向量投影是 (v·u)/(u·u) 乘以 u，余量与 u 正交。
For w equals 3, 4 onto u equals 1, 0, the vector projection is 3, 0 and the orthogonal remainder is 0, 4.
把 w = (3, 4) 投影到 u = (1, 0)，向量投影是 (3, 0)，正交余量是 (0, 4)。
u cross v is perpendicular to both, with length equal to the parallelogram area, so a triangle has half of it.
u × v 与两者都垂直，其长度等于平行四边形面积，所以三角形面积是它的一半。
Reversing the order reverses the vector but keeps the area.
交换次序使向量反向，但面积不变。
Build both edge vectors from the same vertex.
两条边向量要从同一个顶点出发。
The distance from a point to a plane divides by the normal's length.
点到平面的距离要除以法向量的长度。
The triple product gives signed volume; zero means dependence.
三重积给出带符号的体积；为零表示线性相关。
For four planar vectors, order their directions around the circle: each gap would have to exceed 90 degrees, and four such gaps exceed 360.
对四个平面向量，把它们的方向绕圆周排序：每个间隔都要大于 90°，四个间隔之和就超过 360°。
So all six products cannot be negative.
所以六个点积不可能都为负。
The edges 2, 0, 0 and 0, 3, 0 have cross product 0, 0, 6, so the triangle has area 3.
边 (2, 0, 0) 与 (0, 3, 0) 的叉积是 (0, 0, 6)，所以三角形面积是 3。
The projection of 3, 4 onto 1, 0 is 3, 0.
(3, 4) 在 (1, 0) 上的投影是 (3, 0)。
The plane 2x minus y plus 2z equals 6 has normal length 3, so its distance from the origin is 2.
平面 2x − y + 2z = 6 的法向量长度为 3，所以它到原点的距离是 2。
A cross-product magnitude is parallelogram area, so halve it for a triangle.
叉积的模是平行四边形面积，所以三角形要取一半。
A zero dot product is algebraic orthogonality, even for a zero vector.
点积为零是代数意义上的正交，零向量也算。
Plane distance must divide by the normal length.
到平面的距离必须除以法向量长度。
v dot u is 14 and u dot u is 5, so the projection is 14 fifths of u.
v·u = 14，u·u = 5，所以投影是 (14/5)u。
The residual has zero dot product with u.
余量与 u 的点积为零。
The cross product is minus 6, minus 2, 3, with norm 7, so the triangle area is 7 over 2.
叉积是 (−6, −2, 3)，模为 7，所以三角形面积是 7/2。
Order the directions around the circle.
把方向绕圆周排序。
Every consecutive gap must exceed a right angle, or that pair would have a nonnegative product.
每个相邻间隔都必须大于直角，否则那一对的点积非负。
Four such gaps exceed a full turn: impossible.
四个这样的间隔之和超过一整圈：不可能。
With nonpositive products, e1, e2, minus e1, minus e2 works.
若允许非正点积，e₁、e₂、−e₁、−e₂ 就可以。
Check yourself.
自我检测。
A negative dot product means an obtuse angle.
点积为负意味着钝角。
The projection of 3, 4 on 1, 0 is 3, 0.
(3, 4) 在 (1, 0) 上的投影是 (3, 0)。
A cross product of length 6 gives triangle area 3.
长度为 6 的叉积给出三角形面积 3。
The plane is distance 2 from the origin.
平面到原点的距离是 2。
And four planar vectors cannot all have negative products.
四个平面向量的点积不可能都为负。