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--- lecture_16 [2017/03/29 15:47] – [Example ($n=4$)] rupert
+++ lecture_16 [2017/03/30 09:20] (current) – [Chapter 3: Vectors and geometry] rupert
@@ Line 89: / Line 89: @@
-====== Chapter 3: Vectors and geometry ======
-Recall that a $2\times 1$ column vector such as $\def\m#1{\begin{bmatrix}#1\end{bmatrix}}\m{4\\3}$ is a pair of numbers written in a column. We are also used to writing points in the plane $\mathbb R^2$ as a pair of numbersl; for example $(4,3)$ is the point obtained by starting from the origin, and moving $4$ units to the right and $3$ units up.
-We think of a (column) vector like $\vec v=\m{4\\3}$ as an instruction to move $4$ units to the right and $3$ units up. This movement is called "translation by $\vec v$".
-=== Examples ===
-The vector $\vec v=\m{4\\3}$ moves:
-  * $(0,0)$ to $(4,3)$
-  * $(-2,6)$ to $(2,9)$
-  * $(x,y)$ to $(x+4,y+3)$.
-It is convenient to not be too fussy about the difference between a point like $(4,3)$ and the vector $\m{4\\3}$. If we agree to write points as column vectors, then we can perform algebra (addition, subtraction, scalar multiplication) as discussed in Chapter 2, using points and column vectors.
-For example, we could rewrite the examples above by saying that $\vec v=\m{4\\3}$ moves:
-  * $\m{0\\0}$ to $\m{0\\0}+\m{4\\3}=\m{4\\3}$
-  * $\m{-2\\6}$ to $\m{-2\\6}+\m{4\\3}=\m{2\\9}$
-  * $\m{x\\y}$ to $\m{x\\y}+\m{4\\3}=\m{x+4\\y+3}$.
-More generally: a column vector $\vec v$ moves a point $\vec x$ to $\vec x+\vec v$.