Vectors connect arrows and lists of numbers

A vector can look like an arrow or a list of numbers. How can both describe the same thing? In this introduction, 3Blue1Brown uses coordinates to move between the pictures: the arrow makes operations visible, and the list makes them calculable. Watch from 00:11.

For a physics student, a vector is an arrow defined by its length and direction. Moving the arrow without changing either property leaves it the same vector. An arrow in a plane is two-dimensional; one in ordinary space is three-dimensional.

For a computer science student, a vector can be an ordered list of numbers. A house, for example, might be represented by its square footage followed by its price. That is a two-dimensional vector because the list has two entries, and their order matters.

The mathematical perspective generalizes both: a vector can be an object for which addition and multiplication by numbers are suitably defined. The video postpones the details of this abstraction. For now, it starts with a concrete picture: an arrow with its tail at the origin of a coordinate system, then translates the arrow into numbers. This choice differs from the freely movable arrows used in the physics picture.

In two dimensions, the horizontal x-axis and vertical y-axis meet at the origin. Choose a unit length and mark it along each axis. The two coordinates of an arrow rooted at the origin give instructions for reaching its tip: the first says how far to move right or left, and the second how far to move up or down. Positive and negative signs select the direction.

03:44View full image

The video writes the pair vertically in brackets to distinguish a vector from a point. Once the coordinate system is chosen, each ordered pair describes exactly one arrow from the origin, and each such arrow has exactly one pair. Watch the coordinate explanation.

The same idea extends to three dimensions: add a z-axis perpendicular to both x and y. An ordered triplet specifies movement along x, then parallel to y, then parallel to z. Each triplet corresponds to one vector in that space, and vice versa.

To add two arrows geometrically, move the second arrow so its tail sits at the first arrow's tip. Draw the sum from the first tail to the second tip. This construction temporarily moves an arrow away from the origin without changing its length or direction.

05:04View full image

The construction makes sense if each arrow represents a movement: taking the first step and then the second has the same overall effect as taking the sum in one step. It extends the number-line idea behind 2 + 5: move two units right, then five more, for a total of seven.

Consider the vectors (1, 2) and (3, −1). Their movements can be regrouped by direction: move 1 + 3 units horizontally and 2 + (−1) vertically. The sum is (4, 1). In list form, this is the general rule: add corresponding components. Watch the example.

Multiplying an arrow by a number scales it. A factor of 2 makes it twice as long; a factor of 1/3 makes it one-third as long. A negative factor also reverses its direction: multiplying by −1.8 flips the arrow and makes its length 1.8 times the original.

07:22View full image

A number acting in this way is called a scalar. In the numerical representation, scalar multiplication means multiplying every component by that number. For example, doubling (1, 2) gives (2, 4). Watch scaling.

Vector addition and scalar multiplication recur throughout linear algebra. The later abstract perspective focuses on those operations without tying vectors to one representation. Here, the practical skill is translating between arrows and lists, whichever one you regard as fundamental.

A data analyst can use geometry to visualize lists of numbers, patterns and the overall effect of an operation. A physicist or graphics programmer can go the other way, describing and manipulating space with numbers a computer can process. The narrator's animation example follows that path: imagine what happens in space, then calculate where pixels should go.

The next chapter introduces span, basis and linear dependence; this video leaves those concepts for later. Watch the concluding explanation.