📏 向量投影计算器Vector Projection Calculator
求向量 a 在向量 b 上的投影向量 proj_b a = (a·b ÷ |b|²) b、标量投影（a·b ÷ |b|）、与 b 垂直的分量（a − 投影）以及两向量夹角，支持 2–10 维。Project vector a onto vector b: the vector projection proj_b a = (a·b ÷ |b|²) b, the scalar projection a·b ÷ |b|, the component of a perpendicular to b (a − projection) and the angle between them, in 2 to 10 dimensions.
使用说明与常见问题How-to & FAQ
「向量投影计算器」是什么？What is the Vector Projection Calculator tool?
求向量 a 在向量 b 上的投影向量 proj_b a = (a·b ÷ |b|²) b、标量投影（a·b ÷ |b|）、与 b 垂直的分量（a − 投影）以及两向量夹角，支持 2–10 维。Project vector a onto vector b: the vector projection proj_b a = (a·b ÷ |b|²) b, the scalar projection a·b ÷ |b|, the component of a perpendicular to b (a − projection) and the angle between them, in 2 to 10 dimensions.
如何使用「向量投影计算器」？How do I use the Vector Projection Calculator tool?
在「向量投影计算器」中，填写「向量 a（被投影）」、「向量 b（投影到它上面）」等参数，结果会随输入实时计算并显示在下方。In the Vector Projection Calculator: Fill in "Vector a (projected)", "Vector b (projected onto)"; results are recalculated instantly as you type.
向量投影和标量投影有什么区别？What is the difference between vector and scalar projection?
标量投影 a·b ÷ |b| 是一个数，表示 a 沿 b 方向的有向长度；向量投影是这个长度乘以 b 的单位向量，结果是与 b 平行的向量。The scalar projection a·b ÷ |b| is a number: the signed length of a along b. The vector projection is that length times b's unit vector, a vector parallel to b.
垂直分量有什么用？What is the perpendicular component for?
a = 投影 + 垂直分量，把 a 分解为与 b 平行和垂直的两部分，物理中常用来分解力，例如斜面上重力的分解。a = projection + perpendicular part, splitting a into pieces parallel and perpendicular to b — used in physics to resolve forces, such as gravity on a slope.
计算精度如何？How precise are the results?
「向量投影计算器」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Vector Projection Calculator uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「向量投影计算器」免费吗？需要注册吗？Is the Vector Projection Calculator free? Do I need an account?
「向量投影计算器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Vector Projection Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.