General Lines

A line in space that passes through a point A (a, b, c) and parallel to vector v = < v₁, v₂, v₃ > has the equation

This is called the vector equation. It can also be written in parametric equations:

x = a + tv₁

y = b + tv₂

z = c + tv₃

where .

The Forming of Equation

How was the equation generated? Let's take a look at the graph below. The vectors u and v are parallel. The vector u falls on the line of our interest. It starts from point A and ends at a moving point B (x, y, z). The length of u is multiples of v, which gives us 

u	= tv
< x-a, y-b, z-c >	= t < v1, v2, v3 >
< x, y, z > - < a, b, c >	= t < v1, v2, v3 >
< x, y, z >	= < a, b, c > +  t < v1, v2, v3 >
r(t)	= r0 + tv

You may move point B and you will see that vector u and v are multiples of each other. If you move point A, the position of the line will be changed. Moving point P (p₁, p₂, p₃) or Q (q₁, q₂, q₃) will change the direction of v. Take note of the equation on top of the graph, 

v = < q₁-p₁, q₂-p₂, q₃-p₃ >.

  

Line Segment

If we want to describe a line segment, we use the same equation as mentioned above, except the limitation of the interval of t. For example, let's say we refer to the graph above, if the line segment AB have the same length with the vector v (P to Q), then . How do we get the interval? Just refer to the video below to get some ideas. 

Betty, Created with GeoGebra, Screencast-O-Matic and Thomas Calculus, Pearson.