Pure 10.1 - Vectors in two and three dimensions

Vectors as directed components

Can the same vector start at two different places? Imagine moving a counter four squares right and three squares up. Starting at the origin takes you to $(4, 3)$; starting at $(10, -2)$ takes you to $(14, 1)$. The endpoints differ, but the instructions are identical.

The movement is described by a column with the horizontal change above the vertical change. It describes the change, rather than the final coordinates.

The arrow in the picture joins the start directly to the finish; the dashed guides show its two components.

[DIAGRAM: asset_slug: p1_10_1_use_vectors_in_two_and_three_dimensions__diagram_01; asset_name: Horizontal and vertical components of a vector; caption: Four units right and three units up give one directed movement; description: Cartesian axes with a vector from the origin to (4, 3), dashed component guides, horizontal component 4 and vertical component 3, and equivalent column and unit-vector notation]

For $\mathbf a=\begin{pmatrix}-2\\5\end{pmatrix}$, the negative first component means two units left, while the positive second component means five units up. Changing just the first sign gives $\begin{pmatrix}2\\5\end{pmatrix}$, which points in a different direction.

Here is a useful surprise: you can draw infinitely many equal arrows. For any starting point $(p, q)$, the arrow ending at $(p+4, q+3)$ still has components $(4, 3)$. Its location changes; its magnitude and direction do not.

Use that distinction to decide whether an arrow's location matters.

Unit-vector notation in two dimensions

A one-unit movement right is $\mathbf i$, and a one-unit movement up is $\mathbf j$. Combining multiples of these basic movements gives another way to write a column vector.

For example, $5\mathbf i-2\mathbf j$ means five units right and two down, so it is $\begin{pmatrix}5\\-2\end{pmatrix}$. The letters identify the directions; their coefficients give the signed amounts.

Keep the components in the same order when you translate.

Change only the sign of the vertical component in the next example. Both directions are now negative, but their positions do not swap.

A column vector contains the numerical components; the unit vectors sit outside it.

Try the same translation with a different horizontal component.

Three-dimensional vectors

A lift can move vertically while a trolley moves across a floor. To record movement through space, add a third perpendicular direction: the $z$-direction, represented by $\mathbf k$.

The drawing is a perspective view. The axes are perpendicular in space, even though their projections on the page are not all at right angles.

[DIAGRAM: asset_slug: p1_10_1_use_vectors_in_two_and_three_dimensions__diagram_02; asset_name: Three perpendicular component directions; caption: The third component records movement parallel to the z-axis; description: Perspective axes x, y and z with a vector 2i plus 3j plus k, a three-step component path and clearly separated labels for movement 2 in x, 3 in y and 1 in z]

The first two coordinates have not changed their jobs; the third is added after them.

Translate one component at a time rather than trying to remember a visual pattern of signs.