Vector

4 min read#linear-algebra

An arrow with magnitude and direction — the basic object that linear algebra adds, scales, and transforms.

Contents

Vector

A vector is a quantity with both magnitude and direction — an arrow. Velocity, force, and displacement are vectors; temperature and mass, which have only size, are not. Once you fix a coordinate frame, an arrow in the plane is captured by two numbers, its components: \vec v = (v_x, v_y), the horizontal and vertical distances from tail to tip. In three dimensions you add a third, and the same arithmetic carries on into any number of dimensions, where the geometric picture fades but the rules do not.

Components and length

The components are the shadows the arrow casts on the axes. Its magnitude — the length of the arrow — comes straight from the Pythagorean theorem:

\lVert \vec v \rVert = \sqrt{v_x^2 + v_y^2}.
(1)

Drag the tip below to see the components and length update live. The two grey legs are v_x and v_y; the arrow is their resultant.

Drag anywhere to move the vector's tip. The grey legs are its components; the readout shows length and angle.

Adding and scaling

Two operations make vectors a vector space, and almost everything else is built from them. To add vectors, place them tip-to-tail and draw the arrow from the first tail to the last tip — equivalently, add componentwise: (a_x + b_x,\ a_y + b_y). To scale a vector, multiply every component by a number c, stretching its length by |c| and flipping its direction if c < 0.

See also