Foundation lesson

Vectors and their components

A vector is an arrow with a length and a direction. Its components are the numbers that describe it along chosen axes. Turn the axes and the components change while the arrow stays the same: a change of description, not a change of the world.

Written for this edition, not translated from Einstein. Editorial review pending.

How can one arrow have different numbers in different coordinate systems?

A velocity, a force and an electric field each have a size and a direction. Drawn as an arrow, such a quantity is a vector. To write it down with numbers, choose axes and ask how far the arrow reaches along each one. Those numbers are its components, written $(v_x, v_y)$ in two dimensions or $(v_x, v_y, v_z)$ in three.

The components belong to the axes as much as to the arrow. A particle moving at 0.6c along x has components $(0.6c, 0)$ . Describe the same motion with axes turned so that the new x axis makes an angle θ with the old one, and the components become $(0.6c\cosθ,\ -0.6c\sinθ)$ . The numbers changed; the particle did not. Its speed, the arrow's length, is the same in both descriptions.

|\mathbf{v}| = \sqrt{v_x^{2} + v_y^{2}}

The length of v is the square root of v x squared plus v y squared.

Physics often splits a vector into two parts with respect to a direction that matters: the component along it and the part across it. For a boost along x, an electric field of 5 units pointing 3 along x and 4 along y has a parallel component of 3 and a perpendicular part of 4. The split matters in §6 of the relativity paper, where the component along the motion, which Einstein writes X, is the same in both frames, while the components across it, Y and Z, change.

Two different operations can both look like new numbers for a vector. Turning the axes, or moving the observer, changes the description and leaves the world as it was. Pushing the particle changes the world itself. The site's instruments keep the two apart: an observer change never starts a new experiment.

Worked example: One velocity in two sets of axes

A particle moves at 0.6c along x. In the first axes its components are $(0.6c, 0)$ .
Turn the axes so that $\cosθ = 0.8$ and $\sinθ = 0.6$ , an angle of about 36.9°.
The new components are $0.6c \times 0.8 = 0.48c$ along the new x axis and $-0.6c \times 0.6 = -0.36c$ along the new y axis.
The length is unchanged: $\sqrt{0.48^2 + 0.36^2} = \sqrt{0.2304 + 0.1296} = \sqrt{0.36} = 0.6$ , so the speed is 0.6c in both descriptions.

JavaScript is off, so the controls below cannot change anything. The construction shows its first setting, and its closing paragraph, “What it shows, in words”, describes the rest.

Try it: one arrow, turned axes

The arrow is fixed: 5 units long, 3 along the original x axis and 4 along y. Turn the axes and read the arrow's components along the new ones.

Angle of the turned axes, in degrees

Type an angle and press Enter, or drag the slider from −90° to 90°. Try 53.13°, where the new x axis lies along the arrow.

In axes turned by 36.87°, the arrow is 4.8 along x′ and 1.4 along y′.

Its length is √(4.8² + 1.4²) = 5, the same 5 as in the original axes, where it is 3 along x and 4 along y.

What it shows, in words

Turning the axes changes the numbers that describe the arrow and leaves the arrow alone. At 36.87° the components are 4.8 and 1.4, at 53.13° they are 5 and 0, and at every angle the length is 5. The figure draws the original axes dashed, the turned axes solid, and dotted lines from the arrow's tip to its component along each turned axis.

Where this lesson stops

A vector is an arrow with a length and a direction. This lesson stops at describing it along different axes.

If you came here from a passage, Back returns you to the exact place you left.

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