explain why a scalar equation of the line exists in 2-D space, but not in 3-D space.
In two-dimensional (2-D) space, we can describe a line using a scalar equation because there is only one degree of freedom - two coordinates (x, y) are needed to uniquely specify a point in this space. We can represent a line in 2-D space using the equation:
y = mx + c
In this equation, m represents the slope of the line, and c represents the y-intercept. By varying the values of m and c, we can describe different lines in 2-D space.
However, in three-dimensional (3-D) space, there are three coordinates (x, y, z) needed to specify a point. Since there is an additional degree of freedom, a single scalar equation is not sufficient to describe a line in 3-D space.
To represent a line in 3-D space, we need to use parametric equations or vector equations. Parametric equations express the position of the points on the line using parameters (such as t or s), while vector equations define the line using a position vector and a direction vector.
For example, a line in 3-D space can be represented using parametric equations like:
x = x₀ + at
y = y₀ + bt
z = z₀ + ct
In these equations, (x₀, y₀, z₀) represents a point on the line, and (a, b, c) represents the direction vector of the line. By varying the values of t, we can obtain different points on the line.
In summary, a scalar equation can describe a line in 2-D space due to the single degree of freedom, but in 3-D space with three coordinates, parametric equations or vector equations are required to represent a line.