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--- lecture_17 [2016/04/05 09:27] – rupert
+++ lecture_17 [2017/03/30 09:21] (current) – rupert
@@ Line 1: / Line 1: @@
+====== 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 $\vec x+\vec v$.
 === Example ===
@@ Line 84: / Line 108: @@
 What about $\vec v+\vec w$? We have $A+\vec v+\vec w=B+\vec w=C$. So $\vec v+\vec w=\vec{AC}$.
-This gives us the triangle law for vector addition: $\vec v$, $\vec w$ and $\vec v+\vec w$ may be arranged to form a triangle:
+This gives us the **triangle law for vector addition**: $\vec v$, $\vec w$ and $\vec v+\vec w$ may be arranged to form a triangle:
 {{ :tri.png?nolink&300 |}}