Implicit Dimensions of Vectors
Given:
- a = 7i + j
- b = i − 4j + k
a alone could represent (7, 1) ∈ ℝ² or (7, 1, 0) ∈ ℝ³. Its dimension is not explicit.
However, b is clearly 3D because it contains a k-component. Since operations such as a · b require vectors of the same dimension, the context implicitly assumes
a = (7, 1, 0) ∈ ℝ³.
Without this implicit assumption, a ∈ ℝ² and b ∈ ℝ³, so their standard dot product (and cross product) would not be defined.
注意看domain，range，x和其他变量的关系以及变化趋势
注意带变量如t和s的方向向量可以改变magnitude，点不行