Calculate 2D and 3D vector arithmetic in your browser. Computes vector magnitudes, unit vectors, dot product (scalar product), cross product (vector product), angle between vectors, and vector projection.
Essential for linear algebra, physics simulations, 3D graphics, and robotics kinematic calculations.
Frequently asked questions
What is the geometric meaning of the dot product?
The dot product u · v = |u||v| cos θ measures how much two vectors point in the same direction. It is zero if and only if the vectors are perpendicular (orthogonal).
What does the cross product represent?
The cross product u × v produces a vector perpendicular to both u and v, whose magnitude equals the area of the parallelogram formed by u and v.
Can the cross product be calculated in 2D?
In 2D, the cross product is treated as a pseudo-scalar (ux·vy - uy·vx), representing the signed 2D area (determinant).
中文说明
二维与三维空间几何向量计算工具：支持向量模长、归一化单位向量、数量积（点乘/内积）、向量积（叉乘/外积）、夹角（弧度与角度制）、投影向量以及空间距离求解。
广泛适用于大学线性代数、大学物理力学与电磁学、计算机三维图形学与机器人运动学计算。
常见问题
向量点乘（数量积）的几何意义是什么？
点乘公式为 u · v = |u||v| cos θ，衡量两向量在同方向上的投影贡献。两非零向量点积为 0 当且仅当两者互相垂直正交。
向量叉乘（向量积）的几何意义是什么？
三维叉乘结果是一个同时垂直于 u 和 v 的法向量，其模长在数值上精确等于以 u 和 v 为邻边构成的平行四边形面积。
二维向量可以做叉乘吗？
在 2D 平面中，叉乘被定义为伪标量 ux·vy − uy·vx（即二阶行列式），表示两向量张成的有向平行四边形面积。