The vector form of a line equation provides a powerful way to represent lines in space. The equation r equals a plus t times v describes how any point on the line can be found by starting at a known point a and moving along the direction vector v by some scalar multiple t.
Let's examine each component of the vector form. The position vector a specifies a known point on the line, here at coordinates 2, 2. The direction vector v determines the line's direction and slope. The parameter t controls movement along the line - when t is negative, we move backward from point a, when t is zero, we're at point a, and when t is positive, we move forward.
To convert two points into vector form, we follow three simple steps. First, choose one point as our position vector a. Here we select P1 at coordinates 1, 1. Second, find the direction vector by subtracting the coordinates: v equals P2 minus P1, which gives us 3, 2. Finally, we write the vector form: r equals 1, 1 plus t times 3, 2. This equation now represents the entire line passing through both points.
直线方程的向量形式是解析几何中的重要概念。它使用向量来表示直线，提供了一种直观且强大的几何描述方式。与传统的直线方程相比，向量形式更容易理解直线的几何性质。
直线的向量形式表示为 r 等于 a 加 t 乘以 v。这里 r 是直线上任意点的位置向量，a 是直线上已知点的位置向量，v 是直线的方向向量，t 是参数。当参数 t 取不同值时，就得到直线上不同的点。
参数 t 具有重要的几何意义。当 t 等于 0 时，我们得到点 a。当 t 大于 0 时，点沿着方向向量 v 的正方向移动。当 t 小于 0 时，点沿着方向向量 v 的负方向移动。参数的绝对值表示从点 a 的距离比例。
向量形式自然地扩展到三维空间。在3D空间中，我们有 r 等于 a 加 t 乘以 v，其中每个向量都有 x、y、z 三个分量。这给出了三个参数方程：x 等于 x₀ 加 t 乘以 vₓ，y 等于 y₀ 加 t 乘以 vᵧ，z 等于 z₀ 加 t 乘以 vᵤ。参数 t 仍然控制在3D空间中沿直线的运动。
The vector form of line equations has numerous practical applications. In computer graphics, it's used for rendering and animation. In physics, it describes motion along straight paths. Engineers use it for trajectory analysis and robotics. The key advantages include intuitive geometric interpretation, easy parameter manipulation, natural extension to three dimensions, and efficient computations. The vector form r equals a plus t times v provides a unified and powerful framework for working with lines in any dimension.