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Table of Contents
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 $\vec v=B-A = \def\m#1{\begin{bmatrix}#1\end{bmatrix}}\m{5\\-4}-\m{-1\\3}=\m{6\\-7}$. 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 $c$.