📚 Equations of Planes in 3D Vector Geometry | 三维向量几何中的平面方程
In AQA A-Level Mathematics, the equation of a plane brings together vector algebra, scalar products and three-dimensional coordinate geometry. You need to move confidently between vector, scalar-product and Cartesian forms, and apply them to intersections, angles and distances.
在 AQA A-Level 数学中，平面方程将向量代数、数量积与三维坐标几何结合在一起。你需要熟练地在向量形式、点法式与笛卡尔形式之间转换，并将它们应用于相交、夹角与距离问题。
1. Vector Form of a Plane | 平面的向量形式
A plane is determined by one fixed point and two independent direction vectors that lie in the plane. If A has position vector a and the plane is parallel to b and c, then every point in the plane has position vector r satisfying the vector equation
一个平面由一个固定点以及平面内两个不共线的方向向量决定。如果点 A 的位置向量为 a，且平面平行于方向向量 b 与 c，则平面内任意一点的位置向量 r 满足向量方程
r = a + λb + μc
The scalars λ and μ can take any real values. Changing λ moves along the direction b, while changing μ moves along c; together they generate the whole plane.
标量 λ 和 μ 可以取任意实数值。改变 λ 会沿 b 方向移动，改变 μ 会沿 c 方向移动；两者共同生成整个平面。
For this form to define a plane, b and c must be non-zero and non-parallel. If they are parallel, the equation only describes a straight line.
要使该形式表示一个平面，b 和 c 必须是非零且不平行的。如果
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