📚 Position Vectors and Vector Equations | 位置向量与向量方程
In coordinate geometry, a position vector is a vector that starts at the origin O and ends at a point P. It is written as OP or r and completely describes the location of P relative to O. Unlike a free vector, which only records direction and magnitude, a position vector always refers to a fixed reference point.
在坐标几何中，位置向量是以原点 O 为起点、以点 P 为终点的向量，记作 OP 或 r。它完整地描述了点 P 相对于原点的位置。与仅表示方向和大小的自由向量不同，位置向量始终以固定参考点为基准。
1. What Is a Position Vector? | 什么是位置向量？
A position vector gives the coordinates of a point relative to the origin. In two dimensions, if P has coordinates (x, y), then its position vector is:
r = x i + y j
In three dimensions, for P(x, y, z), the position vector is:
r = x i + y j + z k
位置向量表示一点相对于原点的坐标。在二维平面中，若点 P 的坐标为 (x, y)，则其位置向量为 r = x i + y j。在三维空间中，对点 P(x, y, z)，位置向量为 r = x i + y j + z k。
2. Representing Vectors in Component Form | 用分量形式表示向量
Vectors can be written as column vectors or in terms of the unit vectors i, j, k. For two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂), the vector from A to B is:
AB = (x₂ – x₁) i + (y₂ – y₁) j + (z₂ – z₁) k
Its magnitude is:
|AB| = √[(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²]
向量可以写成列向量或单位向量 i、j、k 的分量形式。对于两点 A(x₁, y₁, z₁) 和 B(x₂, y₂, z₂)，从 A 到 B 的向量为 AB = (x₂ – x₁) i + (y₂ – y₁) j + (z₂ – z₁) k。其模长为 |AB| = √[(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²]。
3. Direction Vector from Two Points | 由两点求方向向量
If a line passes through two fixed points A and B, then a direction vector for the line is simply the vector from A to B. Denote the position vectors of A and B by a and b respectively. Then the direction vector is:
d = AB = b – a
This direction vector tells us the slope and orientation of the line. The vector equation of the line through A and B then becomes:
r = a + t(b – a)
如果一条直线经过两个定点 A 和 B，那么这条直线的一个方向向量就是从 A 到 B 的向量。设 A、B 的位置向量分别为 a 和 b，则方向向量为 d = AB = b – a。这个方向向量决定了直线的斜率和方向。因此，通过 A、B 两点的直线的向量方程为 r = a + t(b – a)。
4. The Vector Equation of a Line | 直线的向量方程
The general vector equation of a straight line is:
r = a + t d
Here a is the position vector of a known point on the line, d is a non-zero direction vector, and t is a scalar parameter. Every real value of t gives one point on the line.
For example, the line through (1, 2) with direction vector (3, 4) has vector equation:
r = (1, 2) + t(3, 4)
直线的一般向量方程为 r = a + t d。其中 a 是直线上某已知点的位置向量，d 是非零方向向量，t 是标量参数。每个实数 t 都对应直线上的一个点。例如，经过点 (1, 2) 且方向向量为 (3, 4) 的直线向量方程为 r = (1, 2) + t(3, 4)。
5. Parametric Equations of a Line | 直线的参数方程
By writing the vector equation in components, we obtain the parametric equations of the line. Let a = (x₀, y₀, z₀) and d = (l, m, n). Then:
x = x₀ + t l
y = y₀ + t m
z = z₀ + t n
These three equations, one for each coordinate, are called the parametric form of the line. They are especially useful when we need to test whether a particular point lies on a line.
将向量方程写成分量形式，就得到直线的参数方程。设 a = (x₀, y₀, z₀)，d = (l, m, n)，则 x = x₀ + t l，y = y₀ + t m，z = z₀ + t n。这三个分别对应各坐标的方程称为直线的参数形式。当我们需要判断某一点是否在直线上时，参数方程特别有用。
6. Converting to Cartesian Form | 转换为笛卡尔形式
To convert the parametric equations into Cartesian form, eliminate the parameter t. If l, m, n are all non-zero, then:
(x – x₀) / l = (y – y₀) / m = (z – z₀) / n = t
In two dimensions, the Cartesian form is simply:
(x – x₀) / l = (y – y₀) / m
If one component of the direction vector is zero, for example l = 0, then the equation becomes x = x₀, and the other two ratios still hold.
要把参数方程转化为笛卡尔形式，需要消去参数 t。若 l、m、n 均不为零，则 (x – x₀) / l = (y – y₀) / m = (z – z₀) / n = t。在二维情况下，笛卡尔形式为 (x – x₀) / l = (y – y₀) / m。如果方向向量的某个分量为零，例如 l = 0，则方程为 x = x₀，其余两个比例仍然成立。
7. Parallel and Perpendicular Lines | 平行与垂直直线
Two lines are parallel if their direction vectors are scalar multiples of each other. That is, if line 1 has direction vector d₁ and line 2 has direction vector d₂, then the lines are parallel when d₁ = k d₂ for some constant k.
两条直线平行当且仅当它们的方向向量互为倍数关系。也就是说，如果直线 1 的方向向量为 d₁，直线 2 的方向向量为 d₂，那么当存在常数 k 使得 d₁ = k d₂ 时，这两条直线平行。
Two lines are perpendicular if their direction vectors have a zero dot product. In 2D, if d₁ = (a₁, b₁) and d₂ = (a₂, b₂), then:
d₁ · d₂ = a₁a₂ + b₁b₂ = 0
两条直线垂直当且仅当它们的方向向量的点积为零。在二维中，若 d₁ = (a₁, b₁)，d₂ = (a₂, b₂)，则 d₁ · d₂ = a₁a₂ + b₁b₂ = 0。
In three dimensions, two lines that are neither parallel nor intersecting are called skew lines. They do not lie in the same plane.
在三维空间中，既不平行也不相交的两条直线称为异面直线。它们不在同一个平面内。
8. Intersection of Two Lines | 两直线的交点
To find the intersection point of two lines, we equate their vector equations. Suppose line 1 is r = a₁ + s d₁ and line 2 is r = a₂ + t d₂. At the intersection, there exist parameters s and t such that:
a₁ + s d₁ = a₂ + t d₂
This vector equation produces two or three scalar equations. Solve two of them for s and t, then substitute into the remaining equation to verify consistency. If the remaining equation is satisfied, the two lines intersect; otherwise they do not.
要求两条直线的交点，需要令它们的向量方程相等。设直线 1 为 r = a₁ + s d₁，直线 2 为 r = a₂ + t d₂。在交点处，存在参数 s 和 t 使 a₁ + s d₁ = a₂ + t d₂。这个向量方程可产生两个或三个标量方程。先由其中两个方程解出 s 和 t，再代入剩余方程检验是否一致。如果剩余方程成立，则两直线相交；否则不相交。
9. Position Vectors in Geometric Problems | 几何问题中的位置向量
Position vectors are powerful tools for solving ratio and collinearity problems. If point P divides the line segment AB in the ratio m : n internally, then the position vector of P is:
p = (n a + m b) / (m + n)
Here a and b are the position vectors of A and B respectively. This formula is often tested in IB examinations. It can be derived directly from the vector equation of the line AB.
位置向量在解决比例和共线问题中非常有用。如果点 P 内分线段 AB，使得 AP : PB = m : n，那么点 P 的位置向量为 p = (n a + m b) / (m + n)。其中 a、b 分别是 A、B 的位置向量。这个公式在 IB 考试中经常出现，可以直接从直线 AB 的向量方程推出。
To check whether three points A, B, C are collinear, it is enough to show that the vectors AB and AC are parallel, i.e. AB = k AC for some scalar k.
要判断三点 A、B、C 是否共线，只需证明向量 AB 与 AC 平行，即存在标量 k 使得 AB = k AC。
10. Common Pitfalls and Exam Tips | 常见易错点与考试技巧
Pitfall 1: Confusing position vector and direction vector. A position vector locates a point; a direction vector gives the orientation of a line. They play very different roles in the equation r = a + t d.
易错点 1：混淆位置向量与方向向量。 位置向量确定一个点的位置，方向向量给出直线的方向。在方程 r = a + t d 中，两者的作用完全不同。
Pitfall 2: Forgetting to check the third equation. When finding the intersection of two lines in 3D, solving only two coordinates may give a false solution. Always substitute the parameters into every equation.
易错点 2：忘记检验第三个方程。 在三维空间中求两直线交点时，只解两个坐标方程可能得到虚假解。一定要将参数代入每一个方程进行检验。
Pitfall 3: Using the wrong point on the line. If you use a different point on the same line, the vector equation looks different but represents the same line. Many students wrongly conclude that two equations are different lines.
易错点 3：在直线上取错了点。 如果取直线上不同的点，向量方程的形式会不同，但表示的同一直线。许多学生因此误判为不同直线。
Tip: Always identify a as a position vector of a known point and d as a direction vector. Write down the parametric equations before attempting to solve coordinate problems.
技巧： 始终先确定 a 是已知点的位置向量，d 是方向向量。在尝试解决坐标问题前，先写出参数方程。
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