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--- lecture_17b [2015/04/01 11:27] – created rupert
+++ lecture_17b [2015/04/02 10:03] (current) – [Examples] rupert
@@ Line 23: / Line 23: @@
 More generally: a column vector $\vec v$ moves a point $\vec x$ to $\vec x+\vec v$.
-==== Example ====
+=== Example ===
 Which vector moves the point $A=(-1,3)$ to $B=(5,-4)$?
-Answer: we need a vector $\vec v$ with $A+\vec v=B$, so $\vec v=B-A = \m{5\\-4}-\m{-1\\3}=\m{6\\-7}$.
+Answer: we need a vector $\vec v$ with $A+\vec v=B$, so $\vec v=B-A = \m{5\\-4}-\m{-1\\3}=\m{6\\-7}$. We write $\vec{AB}=\m{6\\-7}$, since this is the vector which moves $A$ to $B$.
+==== Definition of $\vec{AB}$ ====
+If $A$ and $B$ are any points in $\mathbb{R}^n$, then the vector $\vec{AB}$ is defined by
+\[ \vec{AB}=B-A\]
+(where on the right hand side, we interpret the points as column vectors so we can subtract them to get a column vector).
+Thus $\vec{AB}$ is the vector which moves the point $A$ to the point $B$.
+=== Example ===
+In $\mathbb{R}^3$, the points $A=(3,-4,5)$ and $B=(11,6,-2)$ have $\vec{AB}=\m{11\\6\\-2}-\m{3\\-4\\5}=\m{8\\10\\-7}$.
+==== The uses of vectors ====
+Vectors are used in geometry and science to represent quantities with both a **magnitude** (size/length) and a **direction**. For example:
+  * displacements (in geometry)
+  * velocities
+  * forces
+Recall that a column vector moves points. Its magnitude, or length, is how far it moves points.
+==== Definition: the length of a vector ====
+If $\vec v=\m{v_1\\v_2\\\vdots\\v_n}$ is a column vector in $\mathbb{R}^n$, then its **magnitude**, or **length**, or **norm**, is the number
+\[ \|\vec v\|=\sqrt{v_1^2+v_2^2+\dots+v_n^2}.\]
+==== Examples ====
+  * $\left\|\m{4\\3}\right\|=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5$
+  * $\left\|\m{1\\0\\-2\\3}\right\|=\sqrt{1^2+0^2+(-2)^2+3^2}=\sqrt{1+0+4+9}=\sqrt{14}$
+==== Exercise ====
+Prove that if $c\in \mathbb{R}$ is a scalar and $\vec v$ is a vector in $\mathbb{R}^n$, then
+\[ \|c\vec v\|=|c|\,\|\vec v\|.\]
+That is, multiplying a vector by a scalar $c$ scales its length by $|c|$, the absolute value of $c$.