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

Remark

$\|\vec{AB}\|$ is the distance from point $A$ to point $B$, since this is the length of vector which takes point $A$ to point $B$.

Examples

• The distance from $A=(1,2)$ to $B=(-3,4)$ is $\|\def\m#1{\begin{bmatrix}#1\end{bmatrix}}\vec{AB}\|=\left\|\m{-3\\4}-\m{1\\2}\right\|=\left\|\m{-4\\2}\right\|=\sqrt{(-4)^2+2^2}=\sqrt{20}=2\sqrt{5}$.
• The length of the main diagonal of the unit cube in $\mathbb{R}^3$ is the distance between $0=(0,0,0)$ and $A=(1,1,1)$, which is $\|\vec{0A}\|=\left\|\m{1\\1\\1}\right\|=\sqrt{1^2+1^2+1^2}=\sqrt3$.

Scalar multiplication and direction

Multiplying a vector by a scalar changes its length, but doesn't change its direction.

Definition: unit vectors

Proposition: finding a unit vector in the same direction as a given vector

If $\vec v$ is a non-zero vector, then $\vec w=\frac1{\|\vec v\|}\vec v$ is a unit vector (in the same direction as $\vec v$).

Proof

Using the formula $\|c\vec v\|=|c|\,\|\vec v\|$ and the fact that $\|\vec v\|>0$, we have \[ \|\vec w\|=\left\|\frac1{\|\vec v\|}\vec v\right\|=\left|\frac1{\|\vec v\|}\right|\,\|\vec v\|=\frac1{\|\vec v\|}\,\|\vec v\| = 1.\] So $\vec w$ is a unit vector, and since it's scalar multiple of $\vec v$, it's in the same direction as $\vec v$. ■

Example

What is unit vector in the same direction as $\vec v=\m{1\\2}$?

We have $\|\vec v\|=\sqrt{1^2+2^2}=\sqrt5$, so the proposition tells us that is $\vec w=\frac1{\|\vec v\|}\vec v = \frac1{\sqrt 5}\vec v=\frac1{\sqrt5}\m{1\\2}=\m{1/\sqrt{5}\\2/\sqrt5}$ is a unit vector in the same direction as $\vec v$.

Addition of vectors

If $\vec v=\vec{AB}$, then $\vec v$ moves $A$ to $B$, so $A+\vec v=B$.

If $\vec w=\vec {BC}$, then $\vec w$ moves $B$ to $C$, so $B+\vec w=C$.

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:

We get another triangle by starting at $A$ and translating first by $\vec w$ and then by $\vec v$; the other side of this triangle is $\vec w+\vec v$. But we know that $\vec v+\vec w=\vec w+\vec v$! So we can put these two triangles together to get the parallelogram law for vector addition: