Explain why a line in three-space cannot be represented by a scalar equation.
In three-space, a line is not able to be represented by a single scalar equation. This is because a scalar equation contains one variable and describes a two-dimensional object, such as a plane, rather than a line in three-space.
To understand why, let's consider the geometry. In three-space, a line can move in any direction and is not confined to a single plane. It has both length and direction, which cannot be accurately captured by a simple scalar equation involving only one variable.
To represent a line in three-space, we usually use a vector equation or parametric equations.
In a vector equation, we define the line using a position vector r₀ as well as a direction vector d: r = r₀ + td, where t is a scalar parameter that allows us to traverse the line.
Alternatively, we can represent a line in three-space using parametric equations. Here, we express each coordinate of a point on the line separately using scalar parameters: x = x₀ + at, y = y₀ + bt, and z = z₀ + ct, where a, b, and c represent the direction ratios of the line, and x₀, y₀, and z₀ represent the coordinates of a specific point on the line.
By using either the vector equation or the parametric equations, we can accurately describe the position and direction of a line in three-space.