📚 Applications of Vectors | 向量的应用
Vectors are one of the most transferable tools in A-level mathematics. They allow you to describe position, movement, force and geometry in a compact form. In AQA A-level mathematics, vector methods appear in pure questions, mechanics problems and increasingly in modelling contexts.
向量是 A-level 数学中迁移性最强的工具之一。它们可以用简洁的形式描述位置、运动、力和几何关系。在 AQA A-level 数学中，向量方法出现在纯数学题、力学题以及越来越多的建模情境中。
1. Position Vectors and Displacement | 位置向量与位移
A position vector points from the origin O to a point P. If P has coordinates (x, y, z), its position vector is p = xi + yj + zk. The displacement vector from point A to point B is found by subtracting the position vectors: AB = b − a.
位置向量从原点 O 指向点 P。如果点 P 的坐标为 (x, y, z)，其位置向量为 p = xi + yj + zk。从点 A 到点 B 的位移向量等于两个位置向量之差：AB = b − a。
For example, if A is (2, 1) and B is (5, 7), then AB = (5 − 2)i + (7 − 1)j = 3i + 6j. Its magnitude is √3² + 6² = √45 = 3√5.
例如，如果 A 为 (2, 1)，B 为 (5, 7)，则 AB = (5 − 2)i + (7 − 1)j = 3i + 6j。其模长为 √3² + 6² = √45 = 3√5。
AB = b − a, |AB| = √((xB − xA)² + (yB − yA)² + (zB − zA)²)
2. Vector Equations of Straight Lines | 直线的向量方程
A line in two or three dimensions can be written as r = a + tb, where a is the position vector of a point on the line and b is a direction vector. The scalar parameter t can take any real value.
二维或三维空间中的直线可以写成 r = a + tb，其中 a 是直线上一点的位置向量，b 是方向向量。参数 t 可以取任意实数。
For two given points A(2, 3) and B(
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