**Vectors**

February 2018

Vectors are defined in *n*dimensions:

x = [5]

h1 = quiver(0,0,5,0, "linewidth", 2.5);

set(h1, "maxheadsize",0.02);

set(h1, "autoscalefactor", true);

xlabel("x");

ylabel("y";

axis([-1 6 -1 1]);

x = [2 3]

h2 = quiver(0,0,2,3, "linewidth", 2.5);

set(h2, "maxheadsize",0.02);

set(h2, "autoscalefactor", true);

xlabel("x");

ylabel("y");

axis([-1 3 -1 4]);

x = [4 3 1]

h3 = quiver3(0,0,0,4,3,1, "linewidth", 2.5);

set(h3, "maxheadsize",0.02);

set(h3, "autoscalefactor", true);

xlabel("x");

ylabel("y");

zlabel("z");

axis([-1 5 -1 4 -1 2]);

And so on, all the way to

x = [x1 x2 x3 ... xn]

.
**The Euclidean norm**gives an intuitive idea about the length of a vector:

||x|| = sqrt(x1^2 + x2^2 + ... + xn^2)

On the real number line this length is simply

|x|

, which in turn makes operations such as subtraction intuitive. For example:
x - y = z

The operation stays linear in two dimensions:

Where a subtraction operation can be visually understood to mean the following: