Question

If you draw a vector on a sheet of paper, how many components are required to describe it? How many components does a vector in real space have? How many components would a vector have in a four-dimensional world?

Short answer

Answer: In a four-dimensional world, a vector would have 4 components.

Step-by-step solution

Definition of a Vector

A vector is a mathematical object that has both a magnitude (size) and a direction. It can be represented graphically as an arrow with a specific length (magnitude) and angle (direction) with respect to some reference point or reference frame (e.g., Cartesian coordinates). The components of a vector are the scalar values that specify its magnitude and direction along specific axis.

Vectors on a Sheet of Paper

A sheet of paper is a two-dimensional space. To describe a vector on a sheet of paper, we need two components: one component for its magnitude and direction along the x-axis, and another component for its magnitude and direction along the y-axis. So, a vector on a sheet of paper requires 2 components to be described completely.

Vectors in Real Space

Real space (or three-dimensional space) is the space we live in and experience daily. To describe a vector in real space, we need three components: one for its magnitude and direction along the x-axis, another for its magnitude and direction along the y-axis, and the third one for its magnitude and direction along the z-axis. Therefore, a vector in real space has 3 components.

Vectors in a Four-dimensional World

In a four-dimensional world, we need to introduce another axis to represent the vectors in addition to the x, y, and z-axis. Let's call this axis the w-axis. To describe a vector in this four-dimensional world, we need four components: one for each axis (x, y, z, and w). So, a vector in a four-dimensional world would have 4 components.