📚 Vectors in Space | 空间中的向量
Vectors in three-dimensional space extend the familiar ideas of direction and magnitude from the plane, forming the backbone of IB Mathematics problems involving lines, planes, and their interactions. Mastery of 3D vectors unlocks elegant solutions to geometric questions in physics, engineering, and further calculus.
三维空间中的向量将平面中方向和大小的概念延伸到立体空间，是 IB 数学中处理直线、平面及其相互关系的核心工具。掌握空间向量能帮助你用简洁的方法解决物理、工程以及进阶微积分中的几何问题。
1. From 2D to 3D | 从二维到三维
A vector in the plane is written as (x, y); adding a third coordinate z gives a vector in space, (x, y, z). This single extra dimension allows us to model the real world — height, depth, or any third measurable quantity — while still using the same algebraic rules.
平面向量可表示为 (x, y)；增加第三个坐标 z 就得到了空间向量 (x, y, z)。仅多出一个维度就能让我们模拟真实世界的高度、深度或任何第三个可测量的量，与此同时我们仍能沿用熟知的代数规则。
The notation extends naturally: a vector v can be expressed in terms of the standard unit vectors i, j, k along the x-, y- and z-axes, so v = xi + yj + zk.
记法自然延伸：向量 v 可借助沿 x 轴、y 轴、z 轴的标准单位向量 i、j、k 表示，即 v = xi + yj + zk。
2. Three-Dimensional Coordinate System | 三维坐标系
We use a right-handed Cartesian system: if the x-axis points right and the y-axis points up, the positive z-axis points out of the page. The coordinates (x, y, z) locate any point uniquely, and the position vector from the origin O to P is OP = (x, y, z).
我们采用右手直角坐标系：若 x 轴向右、y 轴向上，则正的 z 轴指向纸面外。坐标 (x, y, z) 唯一确定空间一点，从原点 O 到 P 的位置向量就是 OP = (x, y, z)。
Distances between points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) follow from the vector AB = (x₂ - x₁, y₂ - y₁, z₂ - z₁), whose length is the 3D Pythagoras extension.
两点 A(x₁, y₁, z₁) 与 B(x₂, y₂, z₂) 的距离由向量 AB = (x₂ - x₁, y₂ - y₁, z₂ - z₁) 给出，其长度就是勾股定理在三维的推广。
3. Magnitude of a Vector | 向量的模
For v = (x, y, z), its magnitude or length is |v| = √x² + y² + z². This quantity is always non-negative and corresponds to the straight-line distance from the tail to the tip of the vector.
对 v = (x, y, z)，其模（长度）为 |v| = √x² + y² + z²。该值始终非负，相当于向量起点到终点的直线距离。
A unit vector in the direction of v is obtained by dividing each component by the magnitude: û = v / |v|, provided |v| ≠ 0. Unit vectors are essential for describing directions independently of length.
沿 v 方向的单位向量可通过每个分量除以模长得到：û = v / |v|，前提是 |v| ≠ 0。单位向量在仅描述方向而不关心大小时极为关键。
4. Addition, Subtraction and Scalar Multiplication | 加减法与数乘
(x₁+x₂, y₁+y₂, z₁+z₂). Geometrically, the sum obeys the parallelogram law in space. Subtraction is just addition of the negative vector.
向量加法按分量进行：(x₁, y₁, z₁) + (x₂, y₂, z₂) = (x₁+x₂, y₁+y₂, z₁+z₂)。几何上，和向量遵循空间的平行四边形法则。减法不过是加上负向量。
Multiplying a vector by a scalar k stretches (or shrinks) its length by |k| and reverses direction if k is negative: kv x, k y, k z). Two non-zero vectors are parallel if one is a scalar multiple of the other.
用标量 k 乘以向量会使长度变为 |k| 倍，若 k 为负则反向：kv = (k x, k y, k z)。两个非零向量平行当且仅当其中一个为另一个的标量倍数。
5. Scalar (Dot) Product | 标量积（点积）
The dot product of a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is a·b = a₁b₁ + a₂b₂ + a₃b₃. Crucially, a·b = |a||b| cos θ, where θ is the angle between the vectors.
向量 a = (a₁, a₂, a₃) 与 b = (b₁, b₂, b₃) 的点积为 a·b = a₁b₁ + a₂b₂ + a₃b₃。重要的是 a·b = |a||b| cos θ，其中 θ 为两向量的夹角。
This gives a quick test for perpendicularity: a·b gonal (and non-zero). The dot product also enables projection and calculation of work in physics.
由此可得垂直性的快速判别：a·b = 0 当且仅当两非零向量正交。点积还支撑着投影运算以及物理学中功的计算。
To find the angle, rearrange: cos θ = (a·b) / (|a||b|). Always check the components carefully; a sign error flips the angle.
求夹角时可直接变形：cos θ = (a·b) / (|a||b|)。务必仔细核对分量，一个符号错误就会翻转角度。
6. Vector (Cross) Product | 向量积（叉积）
The cross product a × b produces a vector perpendicular to both a and b. Its components can be calculated using the determinant pattern:
叉积 a × b 给出一个同时垂直于 a 和 b 的向量。其分量可用行列式方式计算：
a × b = (a₂b₃ - a₃b₂) i + (a₃b₁ - a₁b₃) j + (a₁b₂ - a₂b₁) k
The magnitude of the cross product is |a × b| = |a||b| sin θ, which equals the area of the parallelogram spanned by a and b. The direction is given by the right-hand rule.
叉积的模为 |a × b| = |a||b| sin θ，它等于以 a 和 b 为边的平行四边形的面积。方向由右手定则确定。
Cross product is anti-commutative: a × b = - (b × a). It is also useful for finding normals to planes and torques.
叉积具有反交换性：a × b = - (b × a)。它在求平面法向量和力矩时非常有用。
7. Equation of a Line in 3D | 空间直线方程
A line in 3D is defined by a point A with position vector a and a direction vector d. Its vector equation is r = a + td, where t is a real parameter.
空间直线由一点 A（位置向量 a）和一个方向向量 d 确定，其向量方程为 r = a + td，其中 t 为实数参数。
Writing d = (l, m, n) and a he symmetric form: (x - x₀)/l = (y - y₀)/m = (z - z₀)/n, provided none of l, m, n is zero.
令 d = (l, m, n)，a = (x₀, y₀, z₀) 可得参数式：x = x₀ + t l, y = y₀ + t m, z = z₀ + t n。消去 t 便得到对称式：(x - x₀)/l = (y - y₀)/m = (z - z₀)/n，前提是 l, m, n 均不为零。
To check if a point lies on a line, see if its coordinates satisfy the symmetric equations for some t. Two lines in space may be parallel, intersecting, or skew (non-parallel, non-intersecting).
要检验一点是否在直线上，只需看其坐标是否对某个 t 满足对称式。空间两直线可能平行、相交或异面（既不平行也不相交）。
8. Equation of a Plane | 平面方程
A plane can be determined by a point A (position vector a) and a normal vector n perpendicular to the plane. The vector equation is r·n = a·n.
平面可由一点 A（位置向量 a）和一个垂直于该平面的法向量 n 确定，其向量方程为 r·n = a·n。
Expanding with n = (a, b, c) gives the Cartesian equation ax + by + cz = d, where d = a·n. This linear equation is the 3D analogue of the line equation in 2D.
将 n = (a, b, c) 代入展开即得直角坐标方程 ax + by + cz = d，其中 d = a·n。这个一次方程是二维直线方程在三维的类比。
To find a normal vector when three non-collinear points A, B, C are given, compute n = AB × AC. Using a point and the normal, the plane equation follows immediately.
给定三个不共线的点 A、B、C 时，可计算法向量 n = AB × AC。再结合一点与法向量，平面方程即可立即写出。
9. Angle Between Two Lines | 两直线夹角
The angle between two lines is defined as the acute angle between their direction vectors d₁ and d₂. Use the dot product: cos θ = |d₁·d₂| / (|d₁||d₂|). The absolute value guarantees the acute angle.
两条直线的夹角定义为它们方向向量 d₁ 和 d₂ 之间的锐角。利用点积：cos θ = |d₁·d₂| / (|d₁||d₂|)。取绝对值是为了确保求得锐角。
When lines are parallel, the direction vectors are scalar multiples. When they are perpendicular, the dot product of their direction vectors is zero.
当直线平行时，方向向量互为标量倍数；当它们垂直时，方向向量的点积为零。
10. Angle Between a Line and a Plane | 直线与平面的夹角
The angle between a line and a plane is complementary to the angle between the line's direction vector d and the plane's normal n. Hence sin φ = |d·n| / (|d||n|), where φ is the angle between the line and the plane.
直线与平面的夹角与直线的方向向量 d 和平面法向量 n 之间的角互为余角。因此 sin φ = |d·n| / (|d||n|)，其中 φ 即为线面角。
This formula arises because when the line is parallel to the plane, d is perpendicular to n, so sin φ = 0; when the line is perpendicular to the plane, d is parallel to n and sin φ = 1.
该公式的原理是：当直线平行于平面时，d 垂直于 n，故 sin φ = 0；当直线垂直于平面时，d 平行于 n 且 sin φ = 1。
11. Angle Between Two Planes | 两平面夹角
The angle between two planes is the angle between their normal vectors n₁ and n₂. Therefore cos θ = |n₁·n₂| / (|n₁||n₂|). Again we take the absolute value to obtain the acute dihedral angle.
两个平面的夹角就是它们法向量 n₁ 与 n₂ 之间的夹角。因此 cos θ = |n₁·n₂| / (|n₁||n₂|)。同样取绝对值以获得锐二面角。
Parallel planes have proportional normals; perpendicular planes satisfy n₁·n₂ = 0. These simple checks make plane geometry in vectors highly efficient.
平行平面的法向量成比例；垂直平面则满足 n₁·n₂ = 0。这些简单的检验使向量方法处理平面几何变得极为高效。
12. Distances and Applications | 距离与应用
The shortest distance from a point P to a line r = a + td is given by |(AP) × d| / |d|, where AP is the vector from any point A on the line to P. This uses the cross product to find the perpendicular component.
点 P 到直线 r = a + td 的最短距离为 |(AP) × d| / |d|，其中 AP 是从直线上任一点 A 指向 P 的向量。该式利用叉积求出垂直分量。
The distance from a point P (position vector p) to a plane r·n = d is |p·n - d| / |n|. This elegant formula is a direct consequence of projection onto the normal.
点 P（位置向量 p）到平面 r·n = d 的距离为 |p·n - d| / |n|。这个简洁的公式正是向法向量投影的直接结果。
Vector methods also solve intersection problems: find where a line meets a plane by substituting the line's parametric form into the plane equation and solving for t. For two planes, the intersection line's direction is n₁ × n₂.
向量方法还能解决相交问题：将直线的参数式代入平面方程，解出 t 即可得到线面交点。两平面的交线方向即为 n₁ × n₂。
Master these techniques and you will handle IB vector questions on lines, planes, angles and distances with confidence. Practice building a habit of drawing a quick sketch and checking with dot or cross products where appropriate.
掌握这些技巧，你就能自信地应对 IB 向量题中关于直线、平面、角度和距离的各种设问。养成快速画草图并适时用点积或叉积检验的习惯将使你如虎添翼。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图