Discover Linear Algebra: A first course in linear algebra

Plot points $P(1,2)$ and $Q(3,-1)$ in the $xy$-plane. Draw an arrow from $P$ to $Q\text{.}$ This arrow is called the directed line segment $\overrightarrow{PQ}\text{.}$

On the same axes you’ve been working on, plot the point $R$ that has the same coordinates as the components of $\overrightarrow{PQ}\text{,}$ and draw the directed line segment $\overrightarrow{OR}$ where $O$ is the origin. What do you notice about this arrow?

This “common” arrow for $\overrightarrow{PQ}$ and $\overrightarrow{OR}$ (and all arrows in the $xy$-plane just like it) is called a vector. Let’s label this vector $\mathbf{v}\text{.}$ Draw another “copy” of $\mathbf{v}$ so that its initial point is $S(-2,1)\text{.}$ What will be the terminal point for this copy of $\mathbf{v}\text{?}$

Draw the vector $\mathbf{u}=(2,3)$ with its initial point at $P(1,1)\text{.}$ Label this vector on your diagram, and label its terminal point as $Q\text{.}$ Now draw the vector $\mathbf{v}=(3,-1)$ with its initial point at $Q\text{.}$ Label this vector on your diagram, and label this second terminal point as $R\text{.}$ Draw in vector $\mathbf{w}$ corresponding to $\overrightarrow{PR}\text{.}$

Compute the components of $\mathbf{w}$ using a $(\mathrm{\Delta }x,\mathrm{\Delta }y)$ calculation between its initial and terminal points in your diagram, similarly to Task 11.1.b.

Looking at the components of $\mathbf{u}$ and $\mathbf{v}\text{,}$ what do you notice about the components of $\mathbf{w}\text{?}$ Based on this, we should call $\mathbf{w}$ the of $\mathbf{u}$ and $\mathbf{v}\text{,}$ and write $\mathbf{w}=\mathbf{u}\phantom{XX}\mathbf{v}\text{.}$

Now work in the reverse order: on the same diagram you’ve been working on, draw $\mathbf{v}$ starting at $P\text{,}$ then draw $\mathbf{u}$ starting at that terminal point. Where did the second terminal point end up? Turn this into a rule for vector algebra: .

What shape do the four vectors on the outside make? And what is $\overrightarrow{PR}$ relative to that shape (geometrically)?

How should you draw the vector $\mathbf{0}=(0,0)\text{?}$

What happens if you draw $\mathbf{0}$ tail-to-head or head-to-tail with another vector (as in Discovery 11.2)? Turn this into a rule for vector algebra: .

Draw a geometric representation of this rule on a set of axes for $\mathbf{v}=(2,1)$ (use the origin as the initial point for $\mathbf{v}$).

What should the components of $-\mathbf{v}$ be?

On the same set of axes as before, draw $-\mathbf{v}$ with its initial point at the origin.

Which of your three vectors represents $\mathbf{w}-\mathbf{u}\text{?}$

Draw in another vector for $\mathbf{u}-\mathbf{w}\text{.}$

What is the point of this activity?

Draw a representative diagram for the vector sum $\mathbf{v}+\mathbf{v}$ using $\mathbf{v}=(2,1)$ (start with the first initial point at the origin). What are the components of this sum vector?

From both the geometry of what you’ve drawn, and the result for the components of the sum vector $\mathbf{v}+\mathbf{v}\text{,}$ do you think it is reasonable to write $2\mathbf{v}$ to mean $\mathbf{v}+\mathbf{v}\text{?}$

Now draw each of the following, and determine their components: $3\mathbf{v}\text{,}$ $-2\mathbf{v}\text{,}$ $\frac{1}{2}\mathbf{v}\text{,}$ $-\frac{5}{4}\mathbf{v}\text{.}$

Using $\mathbf{u}=(1,1,0)$ and $\mathbf{v}=(1,-1,2)\text{,}$ draw the following on a set of $xyz$-axes: $\mathbf{u}\text{,}$ $\mathbf{v}\text{,}$ $\mathbf{u}+\mathbf{v}\text{,}$ $-\mathbf{u}\text{,}$ $\mathbf{v}-\mathbf{u}\text{,}$ $2\mathbf{v}\text{.}$

Now compute the components of each of the vectors from the previous part of this activity.

Does each rule of vector algebra that we’ve discovered today have a counterpart rule in matrix algebra?

Will the same be true for the algebra of higher-dimensional vectors? (That is, if we consider using $n\times 1$ column matrices to represent $n$-dimensional vectors?)