How do you represent a vector in three-dimensional space?

A vector in three-dimensional space is represented as an ordered triple of numbers (x, y, z).

In more detail, a vector in three-dimensional space is a geometric object that has both a direction and a magnitude (or length). It is typically represented as an ordered triple of numbers (x, y, z), where x, y, and z are the components of the vector along the x, y, and z axes, respectively. These components can be thought of as the distances the vector travels along each axis from the origin (0,0,0) to its endpoint.

The vector is often visualised as an arrow starting from the origin and pointing towards the endpoint. The length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector. The magnitude of a vector can be calculated using the Pythagorean theorem, which in three dimensions is √(x² + y² + z²).

Vectors are fundamental in physics as they are used to represent quantities that have both magnitude and direction, such as force, velocity, and displacement. For example, a force of 5 Newtons acting towards the east would be represented as a vector with a magnitude of 5 and a direction of east.

In addition to their geometric interpretation, vectors in three-dimensional space can also be represented algebraically. This is often done using unit vectors, which are vectors of length 1 along the x, y, and z axes. Any vector can be expressed as a linear combination of these unit vectors. For example, the vector (3, 2, 1) can be written as 3i + 2j + 1k, where i, j, and k are the unit vectors along the x, y, and z axes, respectively.

In conclusion, a vector in three-dimensional space is a powerful tool for representing quantities that have both magnitude and direction. It can be represented geometrically as an arrow, or algebraically as an ordered triple of numbers or a linear combination of unit vectors.