Solving Geometric Problems with Vectors | 向量法解几何问题

📚 Solving Geometric Problems with Vectors | 向量法解几何问题

Vectors provide a powerful and systematic approach to solving geometric problems. By converting geometric relationships into algebraic operations, vectors enable us to handle lines, planes, distances, angles and proofs with clarity and precision.

向量为解决几何问题提供了强大而系统的方法。通过将几何关系转化为代数运算，向量使我们能够清晰、精确地处理直线、平面、距离、角度及几何证明。

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1. Vector Basics and Position Vectors | 向量基础与位置向量

In vector geometry, a point is often identified by its position vector relative to an origin O. If point A has position vector a and point B has position vector b, then the vector from A to B is AB = b – a. This simple relation is the foundation for many geometric constructions.

在向量几何中，一个点通常用相对于原点 O 的位置向量来表示。若点 A 的位置向量为 a，点 B 的位置向量为 b，则从 A 到 B 的向量为 AB = b – a。这个简单关系是许多几何构造的基础。

• An equal vector has the same direction and magnitude as another vector, even if it is translated to a different starting point.

  相等向量与另一个向量具有相同的方向和大小，即使它被平移到不同的起点。

• A zero vector has magnitude zero and no specific direction.

  零向量的大小为零，没有确定的方向。

• The negative of vector a, written –a, has the same magnitude but opposite direction.

  向量 a 的负向量记为 –a，其大小相同但方向相反。

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2. Vector Addition, Subtraction and Scalar Multiplication | 向量的加减与数乘

Vectors are added by the triangle law: if AB = a and BC = b, then AC = a + b. Subtraction is equivalent to adding the negative vector, so a – b = a + (-b).

向量按三角形法则相加：若 AB = a，BC = b，则 AC = a + b。减法等价于加上相反向量，因此 a – b = a + (-b)。

Scalar multiplication λa stretches, shrinks or reverses the direction of a. Two non-zero vectors a and b are parallel if and only if a = λb for some scalar λ.

数乘 λa 会拉伸、缩短或反转 a 的方向。两个非零向量 a 与 b 平行的充分必要条件是存在标量 λ 使得 a = λb。

• The parallelogram law: the sum of two vectors can also be found as the diagonal of the parallelogram formed by them.

  平行四边形法则：两个向量的和也可以表示为以它们为邻边构成的平行四边形的对角线。

• If a = (x1, y1, z1) and b = (x2, y2, z2), then a + b = (x1 + x2, y1 + y2, z1 + z2).

  若 a = (x1, y1, z1)，b = (x2, y2, z2)，则 a + b = (x1 + x2, y1 + y2, z1 + z2)。

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3. Dot Product and Angles | 点积与夹角

The dot product of two vectors a and b is defined by

两个向量 a 与 b 的点积定义为

a · b = |a||b| cos θ

where θ is the angle between the vectors. In Cartesian coordinates, if a = (a1, a2, a3) and b = (b1, b2, b3), then

其中 θ 是两向量之间的夹角。在直角坐标下，若 a = (a1, a2, a3)，b = (b1, b2, b3)，则

a · b = a1b1 + a2b2 + a3b3

Thus the cosine of the angle between the vectors is

因此两向量夹角的余弦为

cos θ = (a · b) / (|a||b|)

• If a · b = 0 and neither vector is zero, then the vectors are perpendicular.

  若 a · b = 0 且两向量均非零，则两向量互相垂直。

• The dot product is commutative and distributive: a · b = b · a and a · (b + c) = a · b + a · c.

  点积满足交换律和分配律：a · b = b · a，a · (b + c) = a · b + a · c。

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4. Cross Product and Normal Vectors | 叉积与法向量

In three dimensions, the cross product a × b produces a vector perpendicular to both a and b. Its magnitude is

在三维空间中，叉积 a × b 产生一个同时垂直于 a 和 b 的向量。其大小为

|a × b| = |a||b| sin θ

If a = (a1, a2, a3) and b = (b1, b2, b3), then

若 a = (a1, a2, a3)，b = (b1, b2, b3)，则

a × b = (a2b3 – a3b2, a3b1 – a1b3, a1b2 – a2b1)

• The direction of a × b follows the right-hand rule.

  叉积 a × b 的方向遵循右手定则。

• If a × b = 0 and neither vector is zero, then the two vectors are parallel.

  若 a × b = 0 且两向量均非零，则两向量平行。

• For a plane containing two non-parallel direction vectors, their cross product gives a normal vector to the plane.

  对包含两个不平行方向向量的平面，这两个向量的叉积给出平面的一个法向量。

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5. Parametric Equation of a Line | 直线的参数方程

A line in space can be described by a fixed point A with position vector a and a direction vector b. The position vector r of any point on the line is

空间中的直线可以由一个定点 A（位置向量为 a）和一个方向向量 b 描述。直线上任意点的位置向量 r 为

r = a + tb, t ∈ ℝ

where t is a scalar parameter. If the line passes through two points A and B, then we may choose b = AB = b – a, giving

其中 t 为标量参数。若直线经过两点 A 和 B，则可取 b = AB = b – a，于是

r = a + t(b – a)

• Each point on the line corresponds to exactly one value of t.

  直线上的每一个点对应唯一的 t 值。

• In 2D, eliminating t gives the familiar Cartesian equation of the line.

  在二维中，消去 t 可以得到常见的直线直角坐标方程。

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6. Equation of a Plane | 平面方程

A plane can be defined by a point A on the plane with position vector a and a normal vector n. A point with position vector <

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