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  1. Linear Algebra
  2. Vectors and Spaces

Vectors

PreviousVectors and SpacesNextLinear combinations and spans

Last updated 5 years ago

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賦予任意坐標方向性，並且可以從任意起點開始

  • Scalar multiplication

    • 視覺上就像一個向量，被直直的拉長 10 倍

v⃗=(2,3)10v⃗=(20,30)\vec{v} = (2,3) \\10\vec{v} = (20, 30)v=(2,3)10v=(20,30)

  • Unit vector

    • A unit vector has a magnitude (or length) of 1.

v⃗=(2,3)unit vector of v⃗=(213, 313)check: (213)2+(313)2=1\begin{aligned} &\vec{v} = \left(2,3\right) \\ &\text{unit vector of } \vec{v} = \left(\frac{2}{\sqrt{13}},\:\frac{3}{\sqrt{13}}\right)\\ \\ &\text{check: }\sqrt{\left(\frac{2}{\sqrt{13}}\right)^2+\left(\frac{3}{\sqrt{13}}\right)^2} = 1 \end{aligned}​v=(2,3)unit vector of v=(13​2​,13​3​)check: (13​2​)2+(13​3​)2​=1​

  • Add & subtract vectors

    • 視覺上就像先往第一個向量前進後，再往第二個向量的 (正或反) 向前進

u⃗=(−5,3)w⃗=(−12,−4)w⃗−u⃗=(−7,−7)\begin{aligned} \vec{u} &= (-5, 3)\\ \vec{w} &= (-12, -4)\\ \vec{w} - \vec{u} &= (-7, -7) \end{aligned}uww−u​=(−5,3)=(−12,−4)=(−7,−7)​