What is a vector? Components, magnitude, and direction

A vector tells us how far to move and in which direction. We can draw that movement as an arrow or write it down using components. Let’s connect those two ways of looking at it.

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Use components to draw a vector

The vector $(3,4)$ moves three units to the right and four units up. If we draw it in standard position, its tail is at the origin and its tip is at $(3,4)$. The length follows from the Pythagorean theorem.

$\sqrt{3^2+4^2}=5$

Write the same components in a row or a column

We can write the components horizontally in a $1\times 2$ row vector or vertically in a $2\times 1$ column vector.

$\left[\begin{array}{rr}3& 4\end{array}\right]$

$\left[\begin{array}{r}3\\ 4\end{array}\right]$

Read the column vectors of a matrix

$M=\left[\begin{array}{rrr}-1& 1& 3\\ 0& 2& -2\end{array}\right]$

Read each column from top to bottom. The column vectors have components $(-1,0)$, $(1,2)$, and $(3,-2)$. Each has two components, so we can draw all three in the same coordinate plane.

The components tell us how far to move horizontally and vertically from the origin.

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