Lines in Space

The position vector of a point P(p₁, p₂, p₃) in space is the vector

P = p₁i + p₂j + p₃k.

X denotes the variable vector

X = xi + yj + zk.

We shall now define the notion of a line in space. The simplest way to describe a line in space is by a vector equation.

The vector equation can also be written as a set of parametric equations x = p₁ + d₁t, y = p₂ + d₂t, z = p₃ + d₃t.

If t is time, the line is the path of a moving particle in space given by these parametric equations.

The three coordinate axes are lines with the following vector equations.

x-axis: X = ti,

y-axis: X = tj,

z-axis: X = tk.

Example 4: Finding the Vector Equation Of a Line

If A is a point on L, let us call A a position vector of L. A vector D is said to be a direction vector of L if D is the vector from one point of L to another point of L. Thus if A and B are distinct position vectors of L, then B - A is a direction vector of L (Figure 10.3.7).

Figure 10.3.7