Vectors: arrows and lists

A small plane points due east and flies through the air at 200 km/h. But the air itself is moving: a wind is blowing toward the north at 50 km/h. Where does the plane actually go?

Not east. Each hour, the plane moves 200 km east through the air, and the air carries it 50 km north. Over the ground, it slides off at an angle. Pilots have to work this out before every flight, and they draw it as a triangle of arrows, the wind triangle: the plane’s arrow, plus the wind’s arrow, gives the arrow of where you really go.

Those arrows are vectors, and they’re the starting point of linear algebra, the math that runs 3D games, search engines, image compression and machine learning. This course is going to be all about moving arrows around. First, let’s get to know a single arrow.

The real words

• A Vector in the plane is a pair of numbers $v = (v_1, v_2)$, its components. Drawn as an arrow, it starts at the origin $(0, 0)$ and ends at the point $(v_1, v_2)$. Mathematicians often write it as a column:

  $$v = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$$

• Adding and scaling vectors:

  • Vector addition adds matching components. In the picture, put the second arrow’s tail on the first arrow’s tip:

    $$\begin{bmatrix} 2 \\ 1 \end{bmatrix} + \begin{bmatrix} 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$$

  • Scalar multiplication multiplies every component by one ordinary number, a scalar. $-1 \cdot v$ flips $v$; $0 \cdot v$ is the zero vector $(0, 0)$.

  • Subtraction is adding a flipped arrow: $w - v = w + (-1)v$, the arrow from $v$‘s tip to $w$‘s tip.

• The length (or norm) comes from Pythagoras, since the components are the two short sides of a right triangle:

  $$|v| = \sqrt{v_1^2 + v_2^2}$$

  For $(3, 4)$: $\sqrt{9 + 16} = 5$.

• Everything works the same with more components. A vector in 3D is $(x, y, z)$; a vector can have 1,000 components, which you can’t draw, but you add, scale and measure it exactly the same way.