Vectors as coordinates and displacements

Source scope: §3.1.1. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Read the same object two ways

The vector $(2,1)$ describes a displacement of two units horizontally and one vertically. As data, the same pair might record two measured features. A real Euclidean space is the set $\mathbb R^n$ of ordered real $n$ -tuples. We use column vectors:

x=(x_1,\ldots,x_n)\ \longleftrightarrow\ \begin{bmatrix}x_1\\\vdots\\x_n\end{bmatrix}\in\mathbb R^{n\times1}.

Coordinates are ordered: $(2,1)$ and $(1,2)$ describe different displacements.

Add, then scale

For $x=(2,1)$ and $y=(-1,2)$ , moving by $x$ and then by $y$ ends at $(1,3)$ . Addition combines corresponding coordinates. Scaling applies the same factor to each coordinate:

x+y=\begin{bmatrix}x_1+y_1\\\vdots\\x_n+y_n\end{bmatrix},\qquad \alpha x=\begin{bmatrix}\alpha x_1\\\vdots\\\alpha x_n\end{bmatrix}.

Thus $1.5(2,1)=(3,1.5)$ and $-2(2,1)=(-4,-2)$ . Positive factors preserve direction; negative factors reverse it. Length scales by $|\alpha|$ . At zero, the output is the zero vector, which has no direction.

The same operations apply in three dimensions and in dimensions too high to draw. Physical space supplies intuition for length, distance, and angle; high-dimensional coordinates retain the algebra. A plane picture illustrates these operations but does not limit their dimension.

Explore it yourself

Build a vector from two ingredients

Predict whether changing the second direction can make every point reachable.

Preparing interactive controls…

Coefficient a1

Coefficient b1

u = (1.000, 0); v = (1.000, 2.000). a u + b v = (2.000, 2.000). The directions are independent. Unrestricted coefficients reach any point in the plane; the sliders show a bounded sample.

Axes: first and second real coordinates, in unit steps. Endpoints and labels identify each vector.

Explore in three dimensions

Explore it yourself

Add displacements in space

Predict the sum. Then change a coefficient through zero and compare the tip-to-tail path.

Preparing interactive controls…

Teaching supplement · interactive 3D projection · one grid interval = one coordinate unit

Change the mathematics

Coefficient a2

Coefficient b1

u = (1.000, 0, 0.500); v = (0, 1.000, -0.500). a u = (2.000, 0, 1.000); b v = (0, 1.000, -0.500); sum = (2.000, 1.000, 0.500). Sum length = 2.291.

The blue segment starts at the red endpoint. Its displacement is b v even though it does not start at zero. Negative coefficients reverse a displacement; zero removes it. Two independent vectors span a plane inside three-dimensional space.

Camera controls · change only the view

Drag the diagram with a mouse, or use these sliders with a keyboard or touch. Projection can make perpendicular directions look oblique; the coordinate readout is unchanged.

Camera azimuth (degrees)35

Camera elevation (degrees)25

Camera zoom0.9

Red: input or first direction. Blue: second direction or reference. Green: result. Amber: residual, normal, or third direction. Labels distinguish overlapping objects.

Transfer

If $x=(2,-1,3)$ , what are $-x$ and $x+(-x)$ ?

Check all three coordinates

$-x=(-2,1,-3)$ , so $x+(-x)=(0,0,0)$ . Negating only the positive entries would not produce an additive inverse.