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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
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 $\vec v=B-A = \m{5\\-4}-\m{-1\\3}=\m{6\\-7}$. We write $\vec{AB}=\m{6\\-7}$, since this is the vector which moves $A$ to $B$.
Definition
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 $\vec{AB}=\m{11\\6\\-2}-\m{3\\-4\\5}$.